No, the circuit must be completed before power is used. Since the bulb filament is burned out, the circuit cannot be completed. You should leave the switch off to change the bulb.
When the light bulb is burned out the resistance is infinity and therefore the current is zero.
The power used is E*I = 120V * 0 = zero watts.
No electricity is not wasted !
Open circuit like a burnt bulb does not spend any electricity.
on the burned bulb is electric current does not occur, so there is no electric current is lost if you turn on the switch
