I don't think it matters in warm weather.  Everything is fully powered by the shorepower connection.
In the winter I make sure it is on.  I depend on the electric option on the Hydro-Hot to keep the motorome plumbing from freezing.  If the shore power fails I want the genset to autostart ASAP, keeping the HydroHot element hot.  Leaving the inverter on helps draw the batteries down quickly enough that the genset will start.  Further, I keep a dehumidifier running in the winter that makes sure the batteries drop far enough to trigger the genset.