Is it dangerous if the ground pin of a 3 prong plug breaks off in the outlet?
It could be bad if other things go wrong as well, or if the circuit/device is miswired. It also prevents you from plugging in other grounded devices, which could be a problem as well. The best thing to do would be to either remove it, or replace the receptacle. 

Turn off the power to the circuit at the breaker, and verify power is off. Then using needle nose pliers (or tweezers), try to extract the pin.  If you're unable to remove it, have the receptacle replaced by a professional.