Depends where your electricity comes from. If you're burning fossil fuels to make electricity, that's only about 40% efficient, so you need to burn 2.5x as much fuel to get the same amount of heat into the house.

It’d be fun to actually calculate this efficiency. My local power is mostly nuclear so I wonder how that works out.

Sure. That has nothing to do with the efficiency of your system though. As far as you are concerned this is about your electricity consumption for the home server vs gas consumption. In that sense resistive heat inside the home is 100% efficient compared to gas furnace; the fuel cost might be lower on the latter.

Sure, it's "equally efficient" if you ignore the inefficient thing that is done outside where you draw the system box, directly in proportion to how much you do it.

Heating my house with a giant diesel-powered radiant heater from across the street is infinitely efficient, too, since I use no power in my house.

If you don’t close the box of the system at some point to isolate the input, efficiency would be meaningless. I think in the context of the original post, suggesting running a server in winter would be a zero-waste endeavor if you need the heat anyway, it is perfectly clear that the input is electricity to your home at a certain $/kWh and gas at a certain $/BTU. Under that premise, it is fair to say that would not be true if you have a heat pump deployed but would be true compared to gas furnace in terms of efficiency (energy consumed for unit of heat), although not necessarily true economically.

Generating 1kWh of heat with electric/resistive is more expensive than gas, which itself is more expensive than a heat pump, based on the cost of fuel to go in

If your grid is fossil fuels burning the fuel directly is more efficient. In all cases a heat pump is more efficient.

I think this is pretty silly either way.

- There's an upstream loss on electricity directly in proportion to how much you use; ignoring this tilts the analysis in favor of electricity.

- You pay more for heat from electricity than gas, in part because of this loss.

You accelerate the climate catastrophe so there's less need for heating in the long run.