Depends.
All the ones I've seen kill the electronic converter sections cold dead with the power switch. I haven't had my big one apart, so not sure where exactly the power switch is wired into the inverter circuitry.
When they are turned on and running, they typically run a cooling fan, so there's a few watts there.
I did have a 2000 watt HF inverter hardwired into my van with no difficulty. I also had a 1250 watt Vector hardwired to my truck, no drain - it would start fine after sitting for weeks.