The issue is one of headspace tolerance with the looser spec going to the 7.62 NATO. If the bolt will close, pressure won't be an issue but as mentioned above, depending on source, the NATO ammo might have a long body that would be fine in a NATO chamber but simply won't chamber in a SAAMI spec .308Win rifle.
Going the other way can present some interesting problems. The .308 uses a higher max chamber pressure, but the real problem is with the brass. The NATO round uses thicker brass to keep from rupturing when fired in a loose chamber. Firing a .308 in the same loose chamber could result in a case head separation as the brass expands to fill the chamber and the thin brass forward of the web is stretched to failure. Fairly unpleasant situation especially in a semi-auto where the bolt could well unlock right about the same time.