What could cause my bathroom exhaust fan to stop working?
It may be unsafe to run the heater without a working fan.

Drop ALL the power to the room and remove the guts from the assembly. Usually the fan will have a short cord with a plug. Power it up and try to get it to spin (don't get shocked or bitten). 

If it is fail you might be able to get a replacement from the manufacturer. IME this is hit or miss; sometimes I end up buying a completely new (same) fan and swap the guts so I don't have to install much. Manufacturers may not sell you replacement parts but they sure will sell you the whole thing...

Don't throw anything away. You may need the squirrel cage off the old fan motor if you can find its equivalent.

Verify the outlet in the housing is working properly.