The old wiring itself shouldn't use any amount of electricity if nothing is plugged into the outlet that the wire is attached to or any lights are not on.
Electricity should only be going through and running up a bill if you're actually using something that requires electricity.
However if the wiring is old enough to be frayed and causing short circuits then the old wiring could be using electricity due to the resistance of the shorting wire.
If the wire is old enough and frayed then it also poses a fire risk and should be replaced.