If you mean in the immediate vicinity, I'm not sure what reason there is for supposing they did not. One thing that can be a little hard to wrap one's head around when talking about historical landscapes is the extent of reforestation there has been in Europe and North America in the past century, so just because there are trees around a particular city now does not mean they were there a hundred years ago, or even two thousand year ago. Adding to this, it is difficult if not impossible to get a really detailed look at what an ancient landscape looked like. Careful soil sampling can yield a general portrait via pollen analysis, which can give an idea of what sort of trees were in a particular area, but it cannot give more than a qualitative idea of how thickly wooded it was.

Another thing worth keeping in mind is that a forest contains an awfully large amount of wood and it is more difficult to get rid of one purely from the use of wood than you might think. The vast majority of deforestation today (about 80%) is driven by changing land use (conversion of woodlands to stock raising and farming) rather than logging or fuel. Which is not to say deforestation did not happen, but without a rapidly expanding population and globalized agricultural economy, we cannot assume that the dynamics of deforestation were the same in antiquity as they are today, and it was probably a lot less than you might imagine. The general view of forest and deforestation in antiquity has swung from the ecological catastrophe models popular in the 1970s that you will still often hear in the popular press today (deforetstaion caused the fall of Rome etc).

As for how Rome was supplied with fuel, the majority of it would have been in the form of charcoal. Charcoal production requires both careful planning and a significant amount of labor, but its product is about one third the weight of wood by volume and burns about twice as efficiently, which increases the amount that can be economically transported as well as how far away supplies can be. As for these supplies, there were carefully managed woodlands that involved the selection of certain tree types, and there was, as there is today, extensive woodlands along the Apennines of central Italy. Adding to all this, there are a number of alternative fuel sources such as olive pits, pomace, and dung that would provide a crucial supplement to available wood stock. Coal and other fossil fuels were used, but it is not well understood how extensively, and is generally considered to ave been a fairly marginal fuel source.

A good, available, if rather technical, article on this is Robin Veal's Wood and Charcoal for Rome, which primarily concerns the fuel supply of Pompeii but gives a good idea of what this might have looked like and the economic dynamics behind it. As you have probably gathered, though, there is a lot of uncertainty here.

EDIT: I had not realized it was online, but you might be even more interested in William Harris' Defining and Detecting Mediterranean Deforestation, 800 BCE-700 CE, which is a bit easier to read and goes over the historiography well. The broad argument s that there was deforestation, but it was localized, and probably less than you might think.

Some factors at work are ownership and scale.

Even from very early on, the lands around cities were not just free-for-all zones. They were owned by somebody, or by the state, or by a religious authority. It was no different for Rome. There was ager publicus, "public land" all around Rome and up and down Italia, but its availability for exploitation in the historical era is quite unclear. You could probably go and get yourself a bundle of wood, if you needed it, but you couldn't bring your mining consortium there and clear-cut a thousand hectares.

In any case, you as a private citizen would not go there to get a load of firewood, even if you could, and this is because of scale. For any given person, from the lowliest vagrant to to the richest senator, the opportunity cost of getting a bundle of logs from some spot outside the walls of the city was much too high. You had to go there, gather or cut it, and drag it back to your house. You had to already own an axe. You probably had to have some friends to help, or slaves. In any case, it was much simpler and much cheaper to acquire the fuel from a distributor, whose whole business revolved around getting wood (or charcoal, or oil for lamps, or anything else you can imagine) from the periphery to the center. In other words, the scale of the movement of the good (any good) was, even in the archaic period, much larger than the individual, or even the family.

And it is scale that usually prevents resources from being consumed by a city from the center outwards. Rome was situated on the Tiber River. Goods were brought by ship to the port at Ostia, and then barged up to the city proper. The organizational machinery of that maritime commerce was relatively simple and long-standing. Let's say I want to build an apartment block in the city (an insula), and it will require 1500 sturdy timber beams. As a contractor or executor of that contract, I'm certainly not going to wander out into the hinterland of Latium looking for wood. I'm going to contact my materials importer, and tell him I need 1500 timber logs. He, in turn, has networks and connections and will find a source of timber cheapest for him to move. This almost certainly won't be from a spot around Rome, paradoxically. It is much more expensive to pull timber down out of the Apennines overland than it is to, say, float timber down the Rhone, load it on ships at Massilia (Marseilles), sail it to Ostia, and then barge it up the Tiber to the distribution point at the docks at the Forum Boarium. From there, it is carted to the building site. This scenario is all about transaction costs and, again, scale. Alpine fir and spruce are found at Pompeii and Herculaneum, for instance, both in furniture and in building materials. (Wilson in Scheidel, 140). And as a side note, the timber traders were called dendrophoroi, a Greek term for "wood-carriers." They were often thought to be religious officials, but now most agree they were a guild of timber procurers. Not surprisingly, we find them commonly in the departures of the Alps, along the Rhone, and on the slopes of the Apennines (as example).

As /u/Tiako just pointed out, a wood log (like in a modern fireplace) was not necessarily the most preferred fuel for fires in the ancient world. If for cooking, charcoal was better, and in a little hibachi like you see on the patio porches of many apartments today. If you just needed light, oil-burning lamps were vastly superior, cheaper, and did not fill your living space with as much soot. Most of Rome's population were quite poor, and the poor did not have fires in their living space. On the scale of the individual, then, the need for or motivation to go out chopping wood was very low. At larger scales, it was more efficient to bring wood in bulk from the most optimal sources, which usually were not measured by their immediate proximity. In imperial times, the baths of the city must have consumed insane amounts of wood for heating the water. Rome had the luxury of import for these needs, but other, smaller or landlocked cities did not. Deforestation there might have been much more problematic (for instance in North Africa, where there were not as many woodlands to begin with).

Wilson has a good discussion of all this, and specifically addresses both dendrophoroi and the points made by /u/Tiako (p.150). That chapter is in Scheidel's edited volume Cambridge Companion to the Roman Economy (Cambridge 2012). For some heavier reading, see also Erdkamp et al, Ownership and Exploitation of Land and Natural Resources in the Roman World (Oxford 2015).

The short answer to your question is, as many users suggested in their now deleted commented (in-depth answers, please, guys), is that Rome and other major ancient cities did in fact consume all or most of the trees nearby for fuel. And yet, people continued to exist there, and people need fuel; so, clearly, something was going on that brought fuel to the city. Since you asked "how," I'll focus on a couple mechanisms: the market and the state. Whether one of those mechanisms or both was at place in a particular historical situation, one thing to recognize is that there is a broader pattern to land use, ecological change, and human labor here, a pattern we describe with the terms "core" and "periphery." Core regions are sites with higher ratios of labor to land, typically cities or other settlements. Peripheries are regions with lower ratios of labor to land, and thus land outside the city. Typically, we see a pattern of decreasing ratios of labor to land as one gets farther and farther from the city. So, immediately outside the city, we might find market gardening of fairly labor-intensive products like fruit and vegetables, grapes and wine, olives and oil, etc. Somewhat further out, we find less labor-intensive like grain, beyond which we might find land uses that are better thought of as "extensive"--ranching, herding, and so on. Forestry fits best into this last category. Relatively little labor is necessary to clear forests and bring the wood into settlements; sure, trees are heavy and chopping them down with axes is hard (trust me), but trees grow slowly and so on a year-to-year basis, it doesn't take much labor relative to the amount of land in a forest. The ideas of core and periphery are laid out more econometrically by Immanuel Wallerstein (World-Systems Analysis: An Introduction, [Duke, 2004]). William Cronon's Nature's Metropolis: Chicago and the Great West (Norton, 1991) pioneered this approach for specifically environmental history, and scholars have been using concepts like core and periphery to describe the environmental relationships of cities and nature through space and time.

With an understanding of core and periphery in mind, one key to remember is the dramatic differences of scale in the ancient world. Uruk, the first "true" city in the fourth (maybe fifth?) millennium BCE, was something like 30,000 people--hardly a town in our contemporary world. It's also important to recognize that the broader human population on the planet was significantly smaller overall. So, the cities we're talking about were smaller, often much smaller than cities today. Further, in the absence of widespread human settlement, forests were much more extensive than they are today. Early agriculture, early human settlement, and early cities did exist in something of a frontier situation. For our purposes, that means it was often the case that a city might grow in a region with forests, but would effectively consume the forests in its early stages of growth, resulting in wood shortages. This was true of the everyday needs for fuel for cooking and industry, and particularly the case for more specific varieties of wood used for particular construction projects. For example, in The Epic of Gilgamesh (c. 2600 BCE), Gilgamesh and Enkidu journey from Uruk, in southern Mesopotamia, to the "Cedar Forest," perhaps as far away as Lebanon. In the story, this journey takes weeks or longer. On arriving in the forest, they find a landscape alive with monkeys, birds, and insects, and one whose air is deeply fragrant from the cedar trees. They find and battle Humbaba, an ogre who guards the forest, and on killing him they immediately begin to chop down trees as quickly as they can. The finest among these trees is brought back to Uruk and fashioned into a tall door for the palace. Now, we could dissect this story in any number of ways, but the underlying fact is that ancient cities certainly had problems with obtaining enough wood.

This seems to have been the case with Rome. John Perlin's A Forest Journey (Countryman, 2005--an odd book, not quite an academic source but one that appears to be of decent quality) describes Rome's relationships to forests in a few ways. One, it appears that Romans at least to some extent placed their aboriginal or mythic pasts in forests. Virgil, for example, suggested the mother of Romulus was named Silvia, related to Silvius, "forest dweller." Juvenal and other writers suggested similar ancient relationships to forests. Now, the question for us here is less whether these writers were correct, strictly speaking, and more about how they, as Romans living in a large-for-its-time core region, imagined previous relationships to their periphery. And it seems to be the case that they widely regarded Rome's past a sylvan one. We know that in its very early days, Romans likely exported wood to other core areas, that all of its houses were made from wood and had shingle roofs, and that even after being sacked by the Gauls in 390 BCE, there were still sufficient forests nearby to quickly rebuild the city. We also know that Rome's growth came by tapping into these forest resources: Lucretius noted in the first century before Christ that the growth of cultivation and the demands of the city "compelled the forest to retreat further and further up the mountains" (quote taken from Perlin, above).

So, how exactly did the growing city of Rome, or other major ancient cities, continue to grow or even survive once the forests in their immediate region were gone? There's a mixture of two mechanisms: the market and the state. In Rome's case, their economy in many ways was effectively a market economy. That is, the production and distribution of goods was regulated by price. So, as the city of Rome grew, demand for wood grew as well, leading to higher prices. Higher prices for wood made it worthwhile to either deliberately cultivate wood in the vicinity of the city or to transport wood from greater distances. In the former case, we know that Roman farmers and even large plantation owners likely kept woodlots or other patches of land specifically to grow commercial species. These could include quicker-growing conifers like pines, or slow-growing hardwoods like oak. And, the market for wood could include not merely planks and beams cut from the trunks of large trees, but everything down to small branches lopped off the lower reaches of a tree. Probably more important, however, was the transportation of wood from elsewhere. Put in theoretical or spatial terms, Rome, as a large city and the heart of an empire, could create a core-periphery relationship on a much larger, almost continental scale. That is, Rome was such a major concentration of people (labor) and demand, and was well-supplied enough with capital, that it could simply pay people to cut down trees in southern France, the Alps, or the Balkan peninsula, and then transport them to Rome. Michael Williams documents this process in southern Europe extensively in his 2005 book Deforesting the Earth: From Prehistory to Global Crisis. He argues that the entire Mediterranean basin became subordinate to Roman cities and that this led to widespread, epochal deforestation, creating over several centuries the Mediterranean landscape we know today, in which actual stands of forest are comparatively rare.

In addition to the market, it also seems that in many cases there was some state control of forests. Perlin suggests that in many cases the Roman state claimed total ownership of forested land, and restricted access to them. This access could be opened up in moments of crisis, as Perlin suggested was done in 390 BCE, specifically to facilitate reconstruction of city. It is also the case more broadly that agrarian societies have a wide range of mechanisms to manage forest resources, often on smaller scales than some form of central state control. Ramachandra Guha and Madhav Gadgil's history of India, This Fissured Land (California, 1992) argues that caste emerged as a mechanism to partition access to and control of ecological resources. Each caste was typically granted a monopoly over some resource--fish in a body of water, wood in a forest, farmland in a region, whatever. And, contrary to the claims of classical economic theory and free market economics today, the "tragedy of the commons" is not really a huge problem historically. If a group of people have a long-term interest in managing a resource like wood, they are perfectly capable of doing so sustainably. So, while Rome did not have caste systems as we might find in ancient South Asia, there's every reason to suspect that between market mechanisms throughout the Mediterranean basin, state control of woodlands, and local management of wood resources, they managed, even if their long-term effect was the broad deforestation of much of southern Europe.

In a different part of the world, but related: When visiting Teotihuacan we were told that there was extensive deforestation in the surrounding hills caused by the goings on in the city. Is there any evidence to support the claim that this deforestation lead to drought, crop failure, and the eventual collapse of the city?

I would like to add to the idea of private ownership and therefore market forces helping to maintain forests. In one of Cicero's letters to Atticus we see evidence of forest ownership by his wife Terrentia.

We have inspected Terentia’s forest. All I can say is that only the oak of Dodona is lacking to make us feel that we own Epirus itself. We shall be at either Formiae or Pompeii around the Kalends. If we are not at Formiae do please come to Pompeii. That will be very agreeable for us and not much out of the way for you.

This type of ownership would limit access to forest resources, preventing unregulated clearing. Furthermore it would provide incentives to the maintenance of forest regions as a renewable source of income.