Furnace runs on battery and propane. Make sure you have plenty of charge on the the battery and plenty of propane in the bottle as you head into the evening.
Yes if off-grid you will need to find a way to charge the battery every day.
If enroute you should have enough battery to get through the night assuming a long day driving. At least two batteries is pretty standard for camping without power.
Going to take some time for a full tank to freeze. 3/4 full tank will not be hurt from freezing a bit overnight as there is room to expand. Actual pipes would be more of a worry. Water heater can stay on propane.