For maximum efficiency (i.e. melted water per used fuel) make sure the following things are always true:
Having water increases the thermal conductivity between the pot and the snow/water. With just snow you have a smaller contact area.
As long as there is both snow and water in the pot, the temperature stays at the lowest possible (i.e. melting temperature in the prevailing conditions). All energy goes into melting snow. When there is only water, energy will go into raising the temperature of the water. This means it is best to remove water and add snow in regular intervals.
Edit regarding having/opening a lid:
As pointed out by Paparazzi the benefits from having a lid when melting snow is probably negligible. At melting temperature water will hardly evaporate at all (the vapor pressure is at only 0.006atm at 0degC while it is at 1atm when boiling).
Unrelated to handling snow/water but still very relevant:
Only based on experience the most important factor for efficiency is something else: Heat exchange between the flames and the pot itself. That is why devices like Jetboil (heat directed around a pot with "heat exchanger") are very fast at boiling water (usually at the expense of weigth to size ratio). For any burner this means use a wind shield. Preferably one that is as high as the pot and with just a bit bigger diameter than the pot. Also lowering the flame will increase efficiency at the expense of speed.