Can you tell us the make and model of the thermostat/wireless system? I think, not knowing the thermostat, that at a minimum you'll need 3 wires. The R terminal at the furnace is the source of the 24 volts, AC. In the old thermostat, you were simply connecting the 24 volts from the red wire (connected to the red terminal at the furnace and thermostat) to the W wire at the thermostat thru the mercury switch and thereby sending the 24 volts to the W terminal at the furnace causing the heating cycle to start.
Your new thermostat may need to be powered by the 24 volts and that will require the addition of a wire from the C terminal at the furnace to the C terminal at the thermostat. Be sure all power is off when doing any wiring.