Would I need to empty the tank, or can I leave the soap in there to harden and re-melt another time? Just trying to understand the workings of this unit.
You do not have to empty the tank. You can leave candle wax, M&P, oils, etc., in it as long as you want to. If it is wax or M&P then having a Ball Valve Heater to heat the ball valve will eliminate the valve plugging up.
You may find that just the heat from the tank eventually will do the job. You can also use a hair dryer or heat gun to warm it up. Take care that you do not damage the valve with too much heat.
Having one of our insulation jackets will save you money in energy and speed up the heating. If just for overnight, the contents might still be semi-liquid and will not take near as long to heat back to working temp.