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What I'm curious to know is how well one of these 2000 W generator/inverters can run a gas furnace.
Again, I would suggest avoiding the gen/inverter combo.
Let's say a blower motor in the furnace uses a 3/4 Hp motor. That's about 600 Watts rounded up. But the surge or starting power may be 1,800 watts. So a 2000 W unit would barely power it for startup.
Also, don't forget that the thermostat circuit runs off another (separate) 120 Volt circuit (120 Volt primary, 24 VAC secondry) to provide the 24Volts for the relays, but is low power.
In addition, the newer furnaces also have another motor that may power the combustion chamber, usually a 1/4 Hp or 1/2 Hp (running power) unit.