Haiti is a recent example: they basically used up all their trees for firewood. And ancient Rome had to import their firewood from northern Africa, which contributed to the barren state of its modern landscape.
If you're using wood as fuel, any wood will do, in sufficient quantities. But sometimes you need a specific kind of wood, and it's quite easy to over-harvest.
For centuries, the longbow was the ranged weapon of choice in Europe. To make a longbow, you needed yew, a wood with very specific properties. The outer wood of the yew tree resists expansion like a spring, while the inner wood resists compression. By carving a bow from a yew tree so that the inside of its curve was made of heartwood, with the bow still a single piece, the strength of the bow - its springiness - could be huge. The other option for achieving this strength is to make a composite bow, put together from multiple pieces of wood - but this is harder and more limited.
This meant that yew trees of sufficient size and the right shape were highly prized, and became over-harvested. The price of yew wood increased over the decades and centuries, and eventually a point was reached when there was simply none to be had.
Crossbows were used to fill the gap, and eventually firearms arrived on the scene, their development possibly encouraged by the lack of an alternative.
To create a mast for a sailing ship, you need a tall, straight tree to carve it from, as the mast needs to be a single piece of wood to withstand the wind. Again, as ships became bigger, all the really tall straight trees in Europe were used up, and countries started to send expeditions to Scandinavia specifically to find trees suitable for masts.
There's a parallel here: wood of a particular kind is so important that it is over-harvested. Again, this eventually ceased to be a problem with the invention of steam engines. Did the lack of easily accessible masts spur on the development of steam-driven ships? Perhaps. Again, early steamships were pretty inefficient, but they did not need big masts, and coal was at the time plentiful.
So to what degree were changes like the introduction of firearms driven by "advancement" and to what degree by necessity? The rudiments of a technology may exist for a long time, but with established alternatives, there is no real pressure to develop them. Gunpowder and cannon had been around for a long time, but an arquebus had so many disadvantages compared to a bow it must have barely seemed worth it: you had to keep the powder dry and the fuse lit, the gun was unreliable and still required a whole industry to make precise gun parts, powder, bullets and cartridges.
We have to hope that something similar is going to happen with peak oil. There may be no technological leap forward, but rather a pressure to finally develop alternatives that already exist. If we are really lucky, in a hundred years we will wonder why we ever bothered with fossil fuels given you can just build safe, clean and powerful pocket thorium reactors or something.
Of course, if we are unlucky, there is no replacement for oil. Guns could replace longbows, and steam could replace sailing. But the inhabitants of Easter Island never recovered from their encounter with peak wood: with no trees to make ocean-going canoes from, and no alternative, they were trapped on their island.