Nope. Pressures are higher in the 5.56; however, shooting 5.56 in a .223 chamber will be operating at the upper end of 5.56 rated pressure (around 60k psi, which is the max pressure for the cartridge, where 5.56 in a 5.56 chamber usually operates closer to 58k psi). Generally this will not lead to safety concerns as the bolt thrust is still well within its safety margin, but you will have slightly increased wear on the throat of the chamber. I wouldn't recommend interchanging in a rifle with a .223 match chamber, but in autoloaders like AR's I wouldn't be concerned about it (most of them have hybrid chambers these days anyway). One other note is while SAAMI considers .223 and 5.56 to be non-interchangeable, CIP considers them the same cartridge, so European rifles marked .223 are built to shoot 5.56.