My inverter has 3 states: ON, OFF, Standby
OFF it draws NO current
Standby it draws SOME current
ON it draws more current.
So the answer to your question is "YES".. However that said.. I have over 500 amp hours of battery, so if the ONLY current draw was the inverter.. it would take over a week of having it turned on with no load to run 'em down.
Loaded.. Well, it is a 2,000 watt inverter (That means it can suck over 180 amps).