New switch?

Yeah, I'm thinking the fan switch is the problem. Apparently, it's a 3-way switch, which I don't understand why it was used as there is no other light or timer connected to the fan. I think I'll replace with a SPST switch.

2 way switch (or is it 30 so you can use a 2nd switch to control the same device. Looks like the switch must be shorted. I see 3 wires. Where does the other go to?

You have a 3-way switch in your photo, so there must be a second switch somewhere that also controls the fan. Just replace the one in the photo with a new 3-way. Most likely that'll fix things.

If it worked before, I see no percentage in trying to re-engineer the wiring to use an (admittedly cheaper) SPST switch.

RESOLVED. We had an electrician come in and he identified a connection to a timer on the furnace. Apparently, if it's turned on it overrides the fan switch and comes on automatically. I imagine it's connected to the light and turns on after the light has been on for X amount of time.
Thanks all for the input!

> timer on the furnace

That's a new one on me. I'd like to know exactly what it does, and what it has to do with the furnace.

So it's connected to the furnace AND somehow timed with the bathroom light AND it has it's own switch?!?

This is a bizarre setup. Did the electrician fix it such that it works in the strange manner you're describing or did he remove everything and the fan is only controlled by the switch now?

Edited to add: Actually, when you mention the 'light' are you referring to some kind of activity light on the furnace? I interpreted it as the bathroom light. Either way, it's still bizarre.