Many people will tell you that .223 Remington and 5.56 NATO are simply different designations for the same thing. I am not one of those people. A 223 Remington cartridge has a peak pressure of approx. 55,000. If your rifle has SAAMI chamber (Sporting Arms and Ammunition Manufacturers' Institute)
it can safely handle this.
A 5.56 NATO cartridge is loaded to higher than maximum SAAMI recommended pressures. Nato pressures are up around 62,000. The chamber, throat, and lead of a NATO chambered gun are designed to handle the higher pressures.
If you rifle has a standard NATO chamber, or the newer Wylde (chamber designed to shoot nato pressure ammo and still maintain the accuracy of a SAAMI chamber)
chamber you can safely shoot any commercial ammo and any NATO spec ammo as well.
If you rifle has a SAAMI chamber, NATO pressure ammo will damage your rifle.