Striker fired autos tend to have a system that catches the striker/firing pin as the slide comes back forward. That would "unstick" the firing pin in most cases, or maybe just stop the slide in the case where the firing pin can't be unstuck.
As RickB points out, if the firing pin is stuck forward in a "controlled feed" gun where the round slips up behind the extractor as it feeds, the round will probably jam during the feeding process and hang up the gun.
In a situation where nothing actually prevents everything from operating normally then a gun with a stuck firing pin has a good chance of setting off the round prematurely--before it fully chambers--resulting in a blown case and possible injury.
But yeah, in some guns, the gun might fire full auto until the mag is empty. If the gun has decent recoil and the malfunction is unexpected, there's a good chance of someone getting hurt or killed as a full-auto pistol can really jump around a surprising amount.