Replace mechanical switch with electrical switch or relay
If the button is normally open (no connection when not pressed) you can solder two wires at the bottom to the switch contacts and feed those to the circuit which needs to control the 'button'. The button itself can remain in place. There is no need to de-solder it. 

You can use a relay which is the simplest and safest method as that will work independent of how your fan controller works. If the button needs to be pressed for a long time and you want to save power a toggling or latching relay is best but they are bit more difficult to control. 







simulate this circuit &ndash; Schematic created using CircuitLab

A transistor or FET may be possible but for that you would need the schematics and details of how the fan controller works.