Depending on the firing pin assembly design, dry firing centerfire guns (rifles or shotguns) MAY damage the mechanism. It's rare in modern firearms, but it's possible. Dry firing over snap caps is always an option. Cheap snap caps can be made by pressing the primer out of a spent case and replacing it with a rubber eraser.
Rimfires, on the other hand, should NEVER be dry fired. Dry firing rimfire firearms can cause damage to the chamber mouth (peening of the rim) OR, more severe of the two, it can cause damage to the firing pin, either peening or dulling of the pin tip, or even cracking or breaking the pin tip (or more rarely, the shaft). At any rate, don't dry fire rimfires.