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.