As other have said, there will be some draw with no load - usually about .5a.
When off via remote switch, there is usually still some draw - reduced from that of no load.
When off at the switch on the inverter it depends - some still have a smaller draw where some do not.
Need to check the specs on your specific model inverter.
Vince and Susan
2011 Tiffin Phaeton 40QTH (Cummins ISC/Freightliner)
Flat towing a modified 2005 Jeep (Rubicon Wrangler)
Previously a 2002 Fleetwood Pace Arrow 37A and a 1995 Safari Trek 2830.