This is a great question. First of all, we have to consider all the things that could no longer be constructed using wood as a main material--log cabins, other types of housing, sheds, etc. These items would need to be made out of other materials or a wood "lookalike".
We also need to consider heat sources for those who live in cold climates and depend on wood for fuel. These individuals would need to depend on other sources--coal, solar energy, propane, kerosene, gasoline, or some other fuel source.
There would also be the hundreds of thousands of species of animals which would be effected by the absence of their homes, food sources, and sources for building nests, burrows, shelters. Many would be extinct which would further disrupt and effect the ecology of the world.
Think of the the flooding and erosion that we would not be able to control for the lack of trees. There is also the issue of more polluted air worldwide since trees make up the bulk of our natural air filtering system.
I'm sure I'm missing something, but that's a good start for now. Of course, these are not in order of importance...just in the order that they came to mind.
Check out the links below for more information on the importance of trees and wood in our society and worldwide. Good Luck!
If there where to be no wood, you're implying that there would no trees. Trees, of course, are essential for maintaining the physical world as we know it. If they were to suddenly disappear, the impact to the ecosphere wood be incalculable -- without them, life would ultimately adapt, but today's species, from microbes to insects to mammals and Man, would die off and be replaced with something more adapted to a treeless, woodless world.