Do live house wires that are capped off still use/cost energy?

My fiance took out a ceiling light, three bathroom vanity lights, and two more unused lights from our new home (older home we are remodeling). He taped and capped them off, but I have two questions:

1. Do the wires still draw or use energy since they are still connected to the breaker box on one end? (They are taped/capped off on the other end where he cut them.)

2. Do they draw or use energy if the switch they are connected to is on the ON or OFF position? (Does it matter which position the switch is set on basically.)

3. Should he cut or disconnect the live (but now useless) wires from the breaker box itself?

NOTE: We have a high electric bill, so I was wondering if all of these 'live' wires were acting like they were on and running light bulbs or using energy.

Thank you in advance,
Teresa

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