The big problem is shooting 5.56 in a .223 chamber and getting it hot. (Like at a carbine training class)
The tighter .223 chamber "can" cause the primers to "pop" out of the case of a 5.56 cartridge and fall into the trigger mechanism. If that gets into the wrong place it can deadline the rifle until you can get that primer out, which might entail taking the trigger group out of the rifle.
There are several examples of this in Pat Roger's and others AAR for classes. They would usually "fix" them with a 5.56 chamber reamer to open the chamber up some. Ned Christiansen makes a reamer for this.
I have never had or read about an issue with anything blowing up or other damage, but having a primer tie up a gun isn't something that you would want to happen in a defensive situation.
Just target shooting or plinking shouldn't really be a problem.
I am no longer participating in gun forums.