the problem with 5.56 vs .223 is chamber dimensions and spec'd pressures.
5.56 chambers are sloppy, because they're designed for maximum reliability in a semi auto firearm. Specifically the throat is elongated to aid in feeding.
a .223 chamber may be significantly tighter.
put a 5.56 round in a tight .223 chamber and bad things may very well happen because of the increased pressures in a tight match chamber.
worst case scenario for .223 in a 5.56 chamber is that the chamber is sloppy and you'll have some blowby past the chamber, or maybe a case neck that splits because the brass can't expand that far, but it's very unlikely to blow up.
none of this takes into account the possibility of an Ackley chamber in .223. Put a 5.56 round in an ackley chamber and you're probably gonna have a bad day.