A 308 in a 30-06 should be survivable in any modern firearm. I've seen 30-30 fired in a 308 chamber and it was very interesting (M98 action, the guy ran four 30-30's through it before the fifth one failed to extract after firing).
The real problem is when you put a 308 into something like a 270 Win. The bullet is way too big for the bore and you have a complete failure. In an M98 action I saw the aftermath of a 308 being fired in a 7mm Rem Mag chamber (how the thing even shot is beyond me) but the action held, blew the fore stock off the rifle and saved one dumb hunters life.
So, as long as the bore is the same (or larger) the rifle and shooter SHOULD be fine (one shooter had "crappy accuracy" from his 300 Win Mags which was traced to 7mm Rem Mag ammo being shot through the rifle).
But, always double check your ammo. Because even if the RIGHT ammo is put into a rifle, if it is out of spec it can blow up in your face. Case in point a Win model 70 where the owner was handloading by necking down 7.62x51 milsurp brass. He necked the brass down but didn't trim it to 243 spec and the thick neck brass "gripped" the bullet tighter than spec and caused a "kaboom" with the same failure as the 308 in 7mm Mag Mauser, the forestock popped off and the rifle was ruined.
Hope this clears some stuff up.
Jimro