The 5.56 has a longer freebore (throat)
This is the same system that Weatherby used to keep from blowing up rifles chambered for his magnums.... the bullets get a pretty good jump before they ever contact the lands which reduces the pressure "spike". However the military didn't do it for pressure reasons, they did it for reliability reasons.
The previous post that mentions the up to 10,000 psi increase in pressure when a 5.56 is fired in a 223 is a "worst case" senario. You'd have to be shooting a tight chambered short throated chamber to approach that number.
Ever check your freebore in your commercial chamber for reloading?? Chances are you've got well over .100" freebore if your ammo is loaded to SAAMI maximum OAL! Bolt guns that I handload for usually end up with the ogive of the bullet just kissing the lands... this gives me the best accuracy in most rifles, BUT you have to work those loads up..
This senario is extremely unlikely from just usuing NATO spec ammo in your bolt gun.. and even the worst case senario, 60,000 psi aint gonna blow up your gun or even damage it...
SAAMI minimum proof pressures for 223 is 90,000 PSI which is still a 50% margin...