I would say probably also, but if it's a trailer (TT or 5th wheel), I don't know why you would. The heater uses the most fuel of all the appliances and depending on how far you are traveling, it could use a lot. It only takes a little bit to heat it up after arrival, especially if you fire it up as soon as you get there, then go do all your setup chores.
We have, but the flame will probably blow out. If it is an older one with a pilot, you won't have heat anymore. Like edjunior said, you are heating an area that you aren't using, and if you wait until you stop, it only takes a short time to warm up. Unless it is incredibly cold out.
Running a furnace while towing a RV would bother me knowing how the burn chamber and the air ducts are setup. But it different when I look at a RV refrigerator which is smaller flame and shielded from the outside wind rather well. Where more furnaces are mounted on the side of the coach and rather large intake and exhaust port that wind can blow into either blowing out the flame all together or disturbing enough that some that typically doesn't get will now. So water heater and furnace are two appliances that I rather leave off till I get to camp.
I'm going to agree here and say that it would probably be okay but why be the test bed for the rest of us when the possibility exists that you might look in your mirror and see why most prefer not to do this.
Very few 5Ws and trailers have generators. So, what Ed said above. Generally, trailers and 5Ws are not air tight so your propane heater would be working overtime and all the hot air would be sucked out by the vacuum created when you are rolling down the road.
We run with the heat on if temperatures are below freezing and the trailer is not winterized. We have enclosed tanks and plumbing and heater ductwork that directs some of the air down to the tanks.
If the temperatures are in the 30s or 40s we will typically stop and turn on the heat about an hour before we reach our destination so that the trailer is reasonably warm when we arrive.
It's important to have a working charge line (providing 12 volt power from the tow vehicle to the trailer) when doing this for any length of time, or the trailer batteries will become depleted. Any winter camping requires keeping a careful eye on propane levels.