Sorry Tiriaq, but it would likely blow the rifle apart, especially the stock.
The case has to hold the pressure. In this instance the 308 case is so much shorter than the 30-06, that the gas pressure would blow out all over the place, taking things like magazines and wood with it.
This is why it is practically impossible to get a 308 in a 30-06 chamber.
Other dangerous situations are a 300 Savage in a 30-06.
I realize that there is an inherent danger factor anytime you try to use the wrong ammo in any rifle, however, I must unfortunately disagree in this instance.
The .308 is a cut down .30-06. Therefore the case is basically the same size all the way to the .308 shoulder. The length of the brass case is long enough to actually reach the .30-06 shoulder. How does this make it practically impossible to chamber? Did you think about that statement?
(You can also theoretically chamber and fire a .270Win in a .30-06, if the chamber were military or just sloppy. The shoulder length is only about .010" greater than the .30-06. The O/S dimensions of the neck is about .308". That allows the extra length of the neck to fit into the chamber, centreing the case the while the bullet is pointed into the rifling. Since the bullet is wa-a-a-ay smaller than the barrel, no pressure issues, I would imagine that accuracy would suck though.)
Back to topic...That is why the .308 fire-formed brass has a slight shoulder at the mouth. Pull the trigger, if the extractor holds the base against the bolt face, then the primer will be struck. The case is centred in the chamber, powder ignites, pressure builds, forces the brass outward to conform to the chamber. bullet starts downrange. Gas won't escape from the chamber unless it is filled. So, once the brass fills the chamber and even makes a shoulder the brass should be pushed back against the bolt face, obturation must, and will, occur. Escaping gases are contained, and gas will be fully directed downrange. No harm, no foul. Worst case with new brass, overworked shoulders. Although I haven't seen case separations, I wouldn't rule them out but with new, never fired brass. Case separations occur when head spacing is too long. And in this case, since the cartridge headspaces on the shoulder, it is REALLY short.
Personal case:
Many years ago, I bought a used rifle. Gewehr Mauser K98, re-tooled after the war. On it was/is stamped 7.62mm. That was it. Guy who sold it to me also gave me some .308 rounds to check it out. I fired all 5 of them, not checking simply working the action, and each fed, fired, and extracted well. When I was done, I picked up my brass, like I am supposed to. I had already bought .308 dies and when I looked at the brass, I knew something was wrong. I had 5 straight walled cases with just a slight shoulder at the mouth.
I took it back to my buddy. He immediately knew that it was chambered to .30-06. He had simply taken the rifle on trade a few weeks before and thought it was a .308, since that is how it is marked. The owner didn't know either, it had been his grandfather's. He wanted a newer looking rifle. I have since used it as a .30-06 for about 12 years now and it works great.
I would not have done this intentionally, but the rifle proved safe and there were no ill effects. Each round went down range.