Is it dangerous if the ground pin of a 3 prong plug breaks off in the outlet?

Assuming the outlet is correctly grounded, this would be no different from touching the outlet mounting screws - harmless.

However, if the outlet were miswired, for example, the outlet ground actually connected to neutral (because there was no ground in the box) and there was a wiring fault disconnecting the circuit's neutral, the neutral prong, as well as the ground, could become hot.

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.

It's not immediately dangerous, but it is bad simply because it is wrong.

And it may be easy to fix, as others have noted. If so, you've probably already fixed it.

If not, (perhaps the prong is deeply inserted in the outlet), then consider:

1. Turn off the power at the circuit breaker
2. Verify the power is off using a lamp in the other outlet socket
3. Remove the prong by what ever means needed pliers? tweezers?

In any case, you should definitely fix this, at the very least, it will be a problem at some point later when you want to use a three prong plug.