There will never be a problem shooting 7.62x51 ammo in a 308 (unless it is bad ammo like some of the Indian surplus on the market a while back).

It is exactly the opposite of the 5.56/223 situation - milsurp 7.62x51 is safe to shoot in commercial rifles, commercial 308 may not be safe to shoot in military rifles (CETMEs, Indian Enfields, etc.).