A thermostat wire that runs between the air conditioner unit and the furnace can become damaged quite easily, since it usually is exposed along the refrigeration lines that run between the evaporator coil above the furnace and the air conditioner unit. If this wire becomes damaged, you will need to replace the entire length of wire. This involves disconnecting the wire at both units and reconnecting a new thermostat wire.
Things You'll Need
Turn the power off to both the furnace and the air conditioning unit. Turn them off from the circuit breaker panel to be safe.
Open up the furnace access panel and locate the terminals the thermostat wire is connected to. The thermostat wire that serves the furnace is the wire that has two individual wires inside, red and white. Unscrew the red wire and the white wire from the terminals and pull the whole wire out of the furnace.
Go outside to the air conditioning unit and open up the access panel. Note that the red and white wires are connected to the air conditioner with wire nuts. Twist off these wire nuts and remove the red wire and the white wire.
Discard the old thermostat wire and run a new one between your furnace and your air conditioner. Generally, the wire is run with the refrigeration piping and is zip-tied to it.
Strip 2 to 3 inches of the outer insulation jacket off the thermostat wire, using wire strippers. This will expose the red wire and the white wire inside. Remove 1/4 inch of insulation from the end of each of these wires. Do this at both the furnace and the air conditioner unit.
Attach the wires at the furnace terminals. The red wire connects to the "R" terminal, and the white wire connects to the "C" terminal. Attach the wires at the air conditioner. It does not matter in which orientation the wires attach to the air conditioner. Connect them with wire nuts to the short wire pigtails on the unit.
Put the access panels back on and turn on the power to the units. Test the air conditioner by turning your thermostat to cooling mode and lower the temperature. The unit should operate properly.
Both the furnace and the air conditioner are high-voltage units. Always make sure the power is off before you open the access panels.