.223 Remington maximum safe chamber pressure is ~ 50,000 psi.
5.56 maximum safe chamber pressure is ~ 60,000 psi.
In addition, shooting 5.56 in a .223 chamber may cause significant increases in pressure due to the difference in chamber dimensions.
Do not shoot 5.56mm NATO-spec ammo in a .223 Remington chamber.
There's a ton of good info on this on the net. Here's just one of the many articles out there:
The Gun Zone -- SAAMI on 5.56 v. .223 Remington
Matt