We also run ours while going down the road. No problems. You do need to be careful when pulling up to fuel pumps. Open flame and gas vapors are an explosive combination. Diesel less so, but if the pumps dispense gas as well as diesel, the guy on the other side of the pump could spill gas and create a real hazard if your frig is running.
Mac the Fire Guy recommends keeping refrigerators OFF when towing because of potential fire issues. But if I remember correctly, the examples he uses aren't about the flame igniting insulation or anything else as much as it's about propane lines getting damaged. I think all the cases he refers to may be on motorhomes.