On traditional Colt-type SAA's, the firing pin bushing in the frame (which is crimped in from the front) will be struck by the hammer-mounted firing pin, as there is no primer to stop forward progress...this can elongate and burr the hole, and even drive the bushing out of the frame.
Ruger-type single actions have a frame mounted firing pin which rarely fails, Ruger guaranteeing the guns safe for dry-fire.
If you'd like to dry-fire a traditional SAA-type gun, simply insert a small strip of leather in the hammer frame recess.