I have heard of some doing that. But I think the general conscious is, most do not.
I don't think it will hurt anything to run the furnace on a 5er while on the road (or a TT), but don't be surprised if you go through an excessive amount on propane.
If yours has a pilot light that needs to be lit, then it probably will not work. Once the thermostat reaches temperature, the furnace will kickoff. Wind from driving will blow out the pilot light. If yours is an electronic ignition, the flame may have difficulty igniting as wind would be blowing propane away too fast.
All you can really do it try it and see what happens. Worst that will happen is the wind will blow out the flame, furnace will attempt to restart 3 times and then shut down.
I for one have never towed with the furnace or water heater on. Always with the refrigerator on, but never with furnace or water heater.