It is allright to leave inverter on full time provided batteries have a source of recharge. NOTE: many of those inverters are "inline" devices and stand down if shore power is preset (would need to see how yours is wired to be sure)
But if you are not connected to shore power, or another source of power (Generator, Main engine running alternator) I'd turn it off when not needed.
Page 2: IF the TV is where the driver can see it (IE: Overhead) it MUST be off when driving... However DVR's can still record. also if the TV is elsewhere (Say behind the driver's seat) It's ok to be on.. If I had co-travelers I might invest in a "headrest" TV for the kiddies, I can't see it, butr well.. There is a window that would let them watch even with the slide in (right behind my head).