could .308 rounds be fired in a 30-06 rifle?
In some cases yes. If the extractor is the type to hold the case, it will fire, but wont be very accurate.
Years ago I was running sniper school for both the NG and Regular Army. The guard was using M1C/Ds, the Reg Army, M21s.
For the M1s we had M72 and for the M21s we had M118, both came in white boxes.
The guys were working on zeros for the rifles and I started hearing complaints about flyer's. Looking on the ground I found brass that looked like 458 cases without the rims.
Some of the Alaska natives got the wrong box of ammo, shooting M118 (308) out of their Garands.
I had to add a guy to the supply point to make sure each guy got the right ammo.
None of the guns were hurt, just some poor accuracy and hurt prides.