Most likely, not much. It'd go pop more than bang, the bullet would probably leave the barrel at a couple hundred FPS and land on the ground, most likely followed by a bunch of partially burnt or even still flaming powder kernals. Smokeless powder, particularly the med to slow burning powders used in the CF rifles, requires a considerable amount of pressure to sustain ignition or it simply blows itself out. Since the .257cal bullet is much smaller than the .308" bore, it'd have almost no resistance to moving down the bore, creating very little chamber pressure. I'd suspect that you'd burn less than half the powder in the case, with most of the propulsive force coming from the primer.

The problem that can occur happens with the next round if you don't identify the problem and check the bore. It's possible that the undersized bullet didn't make it out of the bore and/or there's a significant amount of unburnt powder in the bore because there wasn't enough pressure to sustain the burn with the loose bullet. If there is and you shoot another correct cartridge, you've got yourself a bore obstruction situation that could be catastrophic.

The real problem doesn't arise until you try to put a slightly large than bore sized projectile through the bore. A great example would be shooting a 7mm (.284") bullet in a .270 Win bore (which is .277"). The difference is only 0.007", so if the .270's throat is generously cut, it may be possible to chamber the round and fire it. The problem is that the .007" too big bullet itself acts as a bore obstruction and the gun could go boom. This is why the .280 Rem has the entire shoulder moved slightly forward, making it impossible to chamber a .280 Rem case in a .270 Win chamber. The 270 Win will chamber in the .280 Rem rifle and fire, but being .007" undersized, it will just skim down the lands harmlessly.

Mike