I only use branches that are not exposed to animal feces or have been lying on the ground. I've haven't worried about insects inside the wood because I cutting live branches (usually--I do have a pile of oak branches sitting in my drive that I use for cages). Once in awhile, I'll feel guilty and use a hose to wash branches in baby cages but not very often.
I have had some kind of insect in some of the bigger branches as noted by the little mounds of sawdust I've found in cages but those bugs stay inside the wood and I haven't found them spreading in my house. One exception was in the Caribbean when some kind of boring insect went into a wood carving (I found a little l pile of sawdust on the dresser) and I just sprayed the bottom with Raid and that was the end of that. I had been using big branches in my parrot cages.