Does it hurt a gun to dryfire it?

Yes. The firing pin is intended to make contact with a cartridge primer. When this doesn't happen, and it "lands" on an empty chamber, this causes resonance in the firing pin which can increase wear and eventually break it. A much better alternative to dry firing would be to use either spent cartridge casings or the "snap cap" dummy cartridges which you can find at most gun shops.

*

Agree with above. In addition, dry firing many rim fire weapons may dent the mouth of the chamber, resulting in misfires.