How can you shoot a gun "too much?" What do you expect to happen?
If it is a bench-rest grade tack driver, after 3,000 rds. or so you'll probably notice your groups opening up and will want to swap in a new barrel, but that's a consumable item that will cost a lot less than the ammo you've put through it by then.
It's going to pick up finish wear from handling it often, but all really loved guns get that. It's a mark of character.
I don't know how many, MANY, rounds you'll have to shoot before the action is shot out. Not even quite sure it's possible.
Shoot it!