Are you going to hack into the remote to wire the relay contacts across the buttons? Do do still want to be able to push the buttons manually to turn on the fan at other times?
I have a way of doing what you want, but it's moderately complicated. You will need access to three wires from the furnace; both sides of the 24Vac control transformer as well as the wire switched by the thermostat. You will need some diodes, filter capacitors, a LM7812 voltage regulator, a CMOS Schmitt-trigger hex inverter, a couple of transistors, and possibly two small 24Vdc relays.
If this is something you want to build, I'll draw up a schematic.