Control a knob on a Honeywell fan via ESP8266
It's likely a Mains Voltage switch. So a relay (or two) of the same specs would work. Since it's two position you have to map it out with a multi-meter. Keep in mind this is Mains voltage, so Danger, beware.

Or if you want to mechanically control it, a stepper motor or solenoid or any other hacked on physical control rod of some type.