Actually *yes im.being that guy*, feeding 5.56 Nato rounds into a rifle chambered for 223 Remington is a good way to destroy your gun and possibly injure your face.
.223 and 5.56 are dimensionally exactly the same specification.
A .223 chambered gun is quite happy with 5.56. NATO loadings. Some very old guns proofed for the 1950s SAMI standard might have problems with 5.56 pressures which wasn't standardised until the 1960s but nothing is going to blow up.
There could at one time be problems the other way though 'Hot' (over NATO spec) loaded .223 rounds could cause extraction problems in a 5.56 as the military chamber usually has slightly 'tweaked' dimensions by a few thou to ensure easy extraction when it's fouled or dirty and to allow for slight variations in ammo tolerance (they have to be able to swallow ammo made anywhere in NATO and possibly in stores for up to 20 years)
A hot loaded .223 case could over expand and require excessive extraction force, this could result in a failure to extract but more usually prolonged use caused excessive wear on the chamber and extractor.
A .223 gun usually has a chamber that conforms to the SAMI standard dimensions and will be a bolt action rifle so it's not a problem. Modern .223 hunting rounds tend to have hotter loading than NATO standard ammunition.
Actual problems were rare and any gun made in the last 30 years is completely safe for either.223 or 5.56.
There is massive exaggerated among the online 'gun community ' who like to pass on and amplify half remembered mythology.
I think a lot of the 'lore' stems from 7.62 NATO and .308 being virtually identical but with some actual subtle dimensional differences that make a big difference.
308 has a marginally shorter headspace and thinner case walls than 7.62 NATO.
This allows 7.62 NATO ammunition to feed reliably and be used in rifles chambered for .308, but can cause .308 Winchester ammunition cases to rupture if used in a rifle chambered for 7.62 NATO.
That's my understanding anyway.
5.56 was just coming in to general issue when I left the Navy. I went on a level one armourer course for the SA80 but never had to work on any.
Most of my experience is with 7.62 .303 and 9mm Para.