For instance did they have planned tree farms where trees were intentionally planted in order to be harvested later? Were there other fuel sources they could switch to? Or would a large urban center have to keep importing lumber from farther and farther away as the trees nearby were cut away?
One area where I've tried to look into this, ancient and medieval copper smelting in Oman, the evidence seems to suggest that they didn't.
With two major caveats: most of the work that I've read on the topic was done in the 1970s and was conducted by a single individual, Gerd Weisgerber, albeit one who pioneered mining archaeology.
So for instance in his lecture Patterns of Early Islamic Metallurgy in Oman, Weisgerber notes:
The production of more than 200,000...tons of [copper] slag would require not less than 280,000 tons of charcoal...which requires 1,400,000 tons of wood. Thinking of the wood for roasting the ores and refining the copper together with household consumption, an estimated sum of two million tons of wood is not too much. If we take 100kg of wood per tree...it would be necessary to cut twenty million trees down. Therefore we might assume that there were forests of a kind in the Oman mountains and promontaries, a fact which should be of some interest for geography.
Cutting some thousands and thousands of trees in a relatively short period of time must have led to real deforestation and desertation, even when the trees grew more densely than today. For the moment, we do not believe in an import of wood for smelting purposes.
We know even from the forest growing conditions of Germany that fuel was the main problem of copper, iron or steel mills in Renaissance and later times. The mills were only to work some months in a year, to save parts of the forest. In a country like Oman, the total cutting of any tree in the mountains surrounding the mining districts (and all over the Oman mountains there are mining districts) means a change to a local drier climate and less perennial flowing water in the wadis. So we think that the desert in the Oman mountains (unlike that of the Ruba al-Khali hinterland [the Arabian "Empty Quarter"]) and promontaries in early Islamic times was man made. Perhaps the lack of fuel caused the end of this copper production boom, or is at least one salient factor. The limited activities 150 years later could use the small renewed and regrown vegetation. If our hypothesis of such a considerable desertation is true, there must have been terrible consequences.
Having driven around those mountains pretty extensively, that is a mind blowing conclusion to me, and yet the sheer size of these slag piles and the barrenness of their surroundings makes it awfully convincing to me.
I'm not entirely sure on what basis Weisgerber concludes that there weren't fuel imports to Oman at the time, but it's perhaps worth noting that the contemporary trade of wood charcoal from the Horn of Africa to Arabia today is leading to exactly the same process of mass-deforestation and desertification.