No, it's a state license. You only have to renew when it's about to expire.
In fact, if you move out of Georgia, your license is still good until it expires. You just can't renew as a non-citizen.
If you are moving within GA and renewal time is at hand, I would go ahead and renew before the move as some people have reported problems in certain counties with them not wanting to issue until you've been a resident of that county for 90 days. It's bogus, but why deal with the hassles if you can renew before you move.