The rawhide backing project on my osage bow turned out pretty bad, and I want to remove it and start over.  I used Titebond III, which I know can be softened with heat, and my thought is to use a heat gun to loosen the rawhide and peel it off.  However, will the heat damage the wood?

No it won't as long as you only get it hot enough to peel off the rawhide.

I have remover hickory backings and rawhide backings(both glued with TB glue) with a heat gun with no damage to the belly wood or the hickory backing. I wasn't worried about hurting the rawhide backing when I removed it.
  Don't concentrate the heat in one place. Get started at one end and once you get the rawhide to release at the end it will peel up with not too much heat and a little effort.

Thanks, Roy and Pat.   I was hoping that would work.