I've not personally run the furnace while traveling but know and read of many that do.
As for heating the R/V prior and traveling, how fast it cools down depends on a lot of factors such as actual ambient temperature and how well your rig is insulated.
Just remember, things don't freeze the instant the temperature reaches freezing. It can take many hours at or below freezing for items to freeze.
2016 Flagstaff 8529IKBS Diamond Package 5th Wheel-Goodyear Endurance Tires
2017 Ford F-Series SCREW 4x4 - Factory Puck B&W Companion
TST Tire Monitor w/Repeater - Sinemate 3500w Gen.
Days Camped: '17=30 ☼ '18=51 ☼ '19=58 ☼ '20=83