If you have a bulb that no longer works and is still in the socket, and the light switch is turned on, does it use electricity (i.e. will your electricity bill go up)?
Thanks,
Jason
No. If the filament is open, the circuit is open and no current flows. If no current flows, no power is consumed based on Ohm’s law as applied to power:
Power = Current × Voltage.
The entire 120 potential will appear across the open filament, but no current will flow.
In short no.
Since you said Bulb, I’m assuming a standard Incandecent(sp) light bulb. If it’s a different type flourecent, Metal halide, Sodium etc… then it does. (Well at least the losses of the transformer use energy)
No. Unless it is a very odd bulb it would be the same as an open circuit. It is safer to leave a “dead” bulb in the socket than to leave the socket open, so just leave it until you choose to replace it.
No, it won’t impact your electric bill - as the other Dopers have posted before me. Using a very, very simple model, you pay only for the electons flowing back to the power company to be re-energized. The electrons cannot flow passed the burned out bulb. In short, no circuit.
It’s the same thing with all the empty sockets (outlets) on your walls at home. There’s “juice” right up to the outlet, but no circuit for return flow back to the power company…just don’t go sticking your fingers in these open outlets! 
Again, this is the “quick and dirty” explanation of the life of an electron!
- Jinx