Depends on the inverter and how he turns it off. I have a 2005 Damon Intruder with (And this is the important part) an AFTER MARKET inverter, Prosine 2.0 by Xantrex.
Now this inverter has a few different ways it can be turned "Off",,, From its remote control/display I can put the inverter in "Standby" mode.
but if I turn the unit off using the on-device switch, or using the remote plugged into the remote switch jack (This is a simple on/off switch, not the remote display/control panel) then it is OFF and no power flows through it.
In my case 12 volt lights will continue to work till the battery goes dead.
PROVIDED..the batteries are connected (in my case the inverter is before the disconnect switch so if you disconnect the batteries 12 volt systems do not work)