Yes. Whether you could say that it was perceived as a problem, I couldn't tell you. However, we have a number of examples of deforestation occurring, and as a result a deteriorating environment. My favorite example, because it is so extreme, is Lebanon.

If you look at the Lebanese flag there is a Cedar tree. But Lebanon, today, is not exactly known for its Cedar production. In Antiquity, however, Lebanese Cedar was renowned as among the very finest materials you could use for ship building, as well as construction. We have evidence of Cedrus lebani (Lebanese Cedar) being used for building materials as early as the Neolithic. Also as fuel in later periods.

The Lebanese Cedar was prized for it's straightness and its growth pattern. By growth pattern I mean that the wood is very compact, which makes it very good for a wide variety of building projects. In Ancient Egypt we see evidence of Lebanese Cedar being used in the construction of among other things sarcophogi. The Ship of Cheops, built in 2900 BC and located under the Cheops pyramid, was built entirely of Lebanese Cedar. In writings from the time of Pharaoh Snerfu, 2600 BC, we have documents detailing 40 ships filled with Lebanese Cedar, and its use in construction and shipmaking. We also have plenty of evidence of its usage in Israel. Sources for this includes the Bible where Cedar is used. This is corroborated by archaeological evidence from sites such as Tell Lachisch (1800 BC). We could in fact spend significant time going in to the popularity of Lebanese Cedar as a building material, but that's not really the question. Do note that in many of these sources, they speak of Phoenician trade, which was a group occupying this region of Lebanon at the time.

We know that Lebanon had lush forests of Cedar at least as early as 2900 BC (earlier, but as far as written sources go the earliest we have is 2900 BC), lasting up until around the Roman Emperor Hadrian (117 AD). But then what happened? Mount Lebanon is a mountain range which covers well over 1/3 of the country, going from the very northern tip to the very south. This region still has remnants of Lebanese Cedar forests, but much of it is composed of arid scrubland. This is where the Cedar groves of Lebanon used to be. But due to the popularity of Lebanese Cedar, much of the Mt. Lebanon area was deforested, leading to significant soil erosion (perhaps not as significant as other activities like quarrying in later periods though). It also led to one of the main exports of Lebanon disappearing. Lebanese Cedar also grows in Turkey and Cyprus, and in clusters throughout the Eastern Mediterranean. So it's not as if the species went extinct. But I'm sure you can imagine the potential issues both ecological and economically for Lebanon after much of its forests had or were being cut down.

Hajar et al. (2010), using pollen data, note a period of deforestation beginning around 8,000 cal BP (mid-Neolithic) in Lebanong, which seems to calm down towards the end of the Bronze Age (~1200 BCish), and then begin again in the Roman period. Interestingly, the beginning of this deforestation in Lebanon may be tied to a period of the same throughout Europe. Ruddiman (2003) has long argued that a rise in Methane and Carbon Dioxide levels are tied to wide-spread deforestation in Europe beginning around 8,000 BP. Rising Methane and Carbon Dioxide levels globally would lead to increased temperatures as these greenhouse gases act as a sort of insulation, causing the earth to better retain heat. Sort of like a blanket.

There is a great deal that can be said about deforestation in antiquity. It's such a well studied topic I struggle to not write an entire thesis on it in response to your question. Though just to mention one more example, read the short article I link at the very end on the Maya, Khmer, and Incas, and how deforestation in those cases may have led to some rough times due to soil erosion and deterioration. I hope this helped answer your question. If you want more information global deforestation I suggest you delve into Ruddiman (2003) as the starting point.

Sauce:
Ruddiman, William F. "The anthropogenic greenhouse era began thousands of years ago." Climatic change 61.3 (2003): 261-293.

Kaplan, Jed O., Kristen M. Krumhardt, and Niklaus Zimmermann. "The prehistoric and preindustrial deforestation of Europe." Quaternary Science Reviews 28.27-28 (2009): 3016-3034.

Kheir, Rania Bou, et al. "Impact of human activities on water-induced soil erosion in the mountainous coastal region of Lebanon." Science et changements planétaires/Sécheresse 12.3 (2001): 157-66.

Hajar, Lara, et al. "Environmental changes in Lebanon during the Holocene: Man vs. climate impacts." Journal of Arid Environments 74.7 (2010): 746-755.

Mikesell, Marvin W. "The deforestation of Mount Lebanon." Geographical Review (1969): 1-28.

Rich, Sara, et al. "Provenancing East Mediterranean cedar wood with the 87Sr/86Sr strontium isotope ratio." Archaeological and Anthropological Sciences 8.3 (2016): 467-476.

Liphshitz, Nili, and Gideon Biger. "Cedar of Lebanon (" Cedrus libani") in Israel during Antiquity." Israel Exploration Journal (1991): 167-175.

Hajar, Lara, et al. "Cedrus libani (A. Rich) distribution in Lebanon: Past, present and future." Comptes Rendus Biologies 333.8 (2010): 622-630.

Yasuda, Yoshinori, Hiroyuki Kitagawa, and Takeshi Nakagawa. "The earliest record of major anthropogenic deforestation in the Ghab Valley, northwest Syria: a palynological study." Quaternary International 73 (2000): 127-136.

Diamond, Jared. "Archaeology: Maya, Khmer and Inca." Nature 461.7263 (2009): 479.