How long can you run an RV furnace on battery?

If there is no other draw on your batteries, you can expect your heater to run continuously for about 12.5 hours.

Can I plug in my camper without a battery?

The answer is no, with a couple of caveats. Most modern power converters will work just fine without a battery present, but, some dont, so its worth testing before you chuck your RV battery. Additionally, without a battery, youre not going to have a backup power supply in case of a power cut.