Heat treating the belly adds compression strength to the limb. With osage, heat treating isn't necessary because it is already strong in compression. I do it anyway because it doesn't hurt anything and it can help. Heat treating works better for whitewoods like hickory, elm, hackberry, maple, etc.
In TBB4, Marc St Louis wrote a chapter on heat treating bow bellies. Marc has studied the affects of heat treating and has come up with the conclusion that it does indeed improve compression strength and performance. Marc builds high performance short selfbows. His method of heat treating is a lot more severe than the way I do it but I still get benefits from it.
Heat treating and heat straightening are two different things with two different effects. Heat straightening, whether wet or dry is used to make corrections like straightening, bending and recurving. With heat straightening you don't want to scorch the wood. With heat treating you want to scorch the belly(only) to increase compression strength but you can also add reflex to the bow at the same times.