Small correction. Off the top of my head i think that the major issue might be a tighter leade on the .223 causing a shorter jump than the .556 is designed for and resulting in unacceptable pressures with 5.56.
Either way: Bad to fire 5.56 in .223.
My impression is that the two main differences between .223 and 5.56 are: 1) the throat on the 5.56 is a bit longer, and 2) the 5.56 produces about a 10,000 psi higher chamber pressure than the .223.
Having said that, quite a few manufacturers seem to have beefed up their chambers for 5.56, even though they are “nominally” chambered for .223. I suppose the longer throat would make a .223 fired from a 5.56 somewhat less accurate, but I’m not an expert.
Of course, you should check with the specific manufacturer, since AR-15’s etc seem to be multiplying like rabbits (NOT apologies to the likes of Carter and Kerry!).
I also think (based on “some” reading) that the 7.62 and the .308 are virtually identical.