The converse problem with putting .308 into a 7.62x51 NATO chamber is too much headspace. It might not fire reliably with the case head not resting properly on the bolt face. Gas may well escape around the shoulder of the brass too.
I reload my own .308 with the 100 gr Speer "Plinker" for punching paper targets. Less recoil, good accuracy and total control of the cartridge dimensions.