Most landowners don't do any type of post sale treatments. Basically, there isn't that great of fire danger when compared with the western states. Dry lightning is pretty rare, here in the East.
I don't know of any fires that were caused by left over topwood . Most foresters prefer to leave the limbwood as a protection from deer browse on young seedlings.
There can be an issue with some insects. A lot depends on when you have a harvest and what is in the residual stand. Peach bark beetle is what causes the gum streak in cherry. They can develop in fresh cherry slash.
Not logging during spring or summer can help limit the problem. You can also remove or cut the limbwood down to firewood size. That will help it dry out better and will lessen nesting sites.