The US Army thought about the 7.62 NATO vs .30-06 compatibility for a long time before adopting the 7.62 x 51 NATO cartridge for service rifles and machine guns. The M1 .30-06 had been the service rifle since 1936, and the M1903 Springfield for several generations before that.
The 7.62 NATO (.308 Winchester in civilian clothes) will chamber tightly in the .30-06 chamber, such that the should dimension interferes slightly in the .30-06 chamber. It will chamber and fire, SAFELY with no case head separation, blown case, or other extreme gas leakage. The US Army tested this concept extensively (At Aberdeen Proving Gournd, I think) in field trials with M1 rifles and M1918 BAR, and M1919 machine guns.
I have personally fired many rounds of .308 Winchester in M1 rifles, Remington M742 rifles, and a few others. All fired with almost full velocity, and no damage to either rifles or shooters. The .308 case blows it's neck out to fill the .30-06 chamber shoulder. Still, no harm done.
In some rifles with tight chambers, the .308 Winchester round will chamber hard, and may require repeated attempts to chamber and fire.
I would not hesitate to fire this combination in a survival or self defense scenario.