If the servo is connected to a power supply it will consume energy even if there is no signal. If there is no load on it the amount of energy consumed will be much smaller.
You could save energy by using the Arduino to control a relay or a transistor to switch off the power supply.
I don't know if it might harm a servo to provide a signal when the servo power is disconnected. If you use servo.detach() the Arduino will stop sending the signal, but I'm not sure if the signal pin will be HIGH or LOW - maybe you would need to set it LOW with digitalWrite().
There can be problems if you power up a servo without a signal. Depending on the model it may move to one or other end of its travel. Connecting a 4k7 resistor between signal and ground might stop that.
Some "suck-it-and-see" is required.
...R