As noted: All outlets in places that get wet (Outside, Sink, Bathroom, over the dining room table) are GFCI protected, usually the GFCI is in the bath, however a popular game at the Factory is Where can we hide a 2nd one.
As for the furnace: AC has nothing to do with it save for this.
The Converter converts 120 volt AC into around 13.6 to 14.6 volts to charge the batteries..
The furnace runs off battery power. IF batteries are low you get blower, but not ignition.
And if you need the full furnace boilerplate (Which describes the operation at every step.. Well,,, I have it, and can post it, but let us start with batteries.
SOME idiots (NOTE: these idiots are at the factory, not speaking to any of the members here) Put the converter on the GFCI loop as well.. And if it tripped, you have no charge.