It sounds like you are talking about the TC battery and not the truck battery. On the average unmodified TC, your running lights, tail lights, backup lights, and turn signals don't connect to the TC battery. They operate from the truck electrical.
One thought, the furnace blower pulls quite a bit of current. I don't have a problem with an overnight furnace run but I have 2 group 29's. How much current do you have available?
The only normal thing I can think of that would run the battery down quickly is if you had a 3 way frig and it was running from 12V. Most are around 300 watts.
But, since you removed the fuses and still ran the battery down, there has to be a path to ground that isn't a dead short.
I think I would check the output connection of the converter. There are filter caps in the output of most converters that could be very leaky and drain the battery without causing a dead short. The converter would still operate when connected to AC power, charging the battery, although the DC output would be noisy.
To verify, depending on the converter, you should be able to pull the fuses on the converter output or disconnect the wires that run from the converter to the fuse box.
I have seen a few flakey Magnatek converters.
Good Luck,
