Only issue with furnace on while traveling is IF floor registers are blocked and/or furnace return air flow is blocked/obstructed.
Furnace needs adequate airflow thru heat exchanger to remove heat generated by propane flame otherwise temp can rise to fast or heat exchanger can get too hot...in either case the 'High Temp Limit Switch' will shut down furnace flame causing furnace to short cycle.
Otherwise run it........
Only restrictions on propane is some tunnels, ferries ----------and they will be CLEARLY MARKED
Otherwise states do NOT prohibit having propane valve open and in use.