Technically, no because a leak is more likely to develop while underway due to roadvibration. A lesser reason is the danger of forgetting to shut off the running apliances before fueling (that should be done before pulling up to the pumps). That said, many, if not most, do it without problems. Personally, unless towing in freezing weather and heat is necessary to prevent freezing the water system, I don't see any real need to run any propane appliances. Unless traveling for a really long time during the day, the refrigerator will stay cold enough during the trip as long as the doors are kept shut. Three-ways can run on 12v while underway and, if run off an inverter, a two-way could run on 120v. As pointed out, water heaters can reheat a tank of water fairly quickly (an hour or so) and furnaces don't need that much time to kill the chill.