Since the .308 has a larger diameter at the shoulder than the .30-'06 does at the same point, most of the time a .308 will not seat in a .30-'06 chamber without force being applied.
When it does seat, and is fired, about all that usually happens is that you have a .308 case with a very short neck. Nevertheless, not recommended, the case could blow at the shoulder rather than just fire forming.
(Yes, been there, done that, as an experiment, not an error.)
Jim