Furnace is a major battery hog - as the battery slowly depletes the furnace fan turns slower and eventually the sail switch within the furnace will close which cuts off the propane to the furnace. The sail switch is a safety feature and kept open by air movement from the fan. If your not connected to shore power and only have one battery this would be high on my list. Also - you might want to check for blown fuse.
BTW - the std battery monitor that comes with most rigs is pretty much worthless - if you have multimeter check the battery voltage.