When there is no signal sent to a servo motor, does it put the circuit in an open state, or still in a closed state? I need to know if I can wired together a servo and a power supply and leave it all connected without the servo draining the battery when not in operation. Thanks.
The servo will draw some current when connected. For example I have a 5g microservo that use about about 5mA, when connected but not doing anything. The current when stalled is about 700 mA
You could use the Arduino to disconnect power to the servo using a relay or suitable transistor.
Power on a stationary servo depends on the torque applied to it - it will fight to maintain the last commanded position (though some cut out eventually). Switching the servo supply off to reduce power drain would probably be worth it. Its worth testing your particular brand of servo to see how it behaves in your set-up.
Remember these things are designed to run for short periods of time from a large battery pack, not long periods of time from a small battery pack, they are not intentionally micro-power.