Controlling an RC car using Arduino
Can you measure how much current goes through the wire when grounded (A: 2 mA from measurements in comments) ? And the open circuit voltage between the controller's ground and the control wire?  (A: 3.1V, from 2xAA)

If the current isn't more than the 40mA per pin/100mA per port, 200mA total, and the open circuit voltage of the controller pins is less than 5V, then you should be able to connect the Arduino's output pins and ground to your controller's wires and ground.  

If the voltages and currents are higher, then you would need protection to prevent damaging your Arduino.    

The controller and the Arduino should share a common ground, but without a circuit and without the controller's voltages and currents, it is hard to tell exactly what is wrong.

Edit: Since the controller's open circuit voltage and short-circuit currents (3.1 V, 2mA max) fit within the Arduino's capabilities, you can wire them directly to the pin outputs, and make sure the controller and the arduino share a common ground. 

You need a fifth wire between the controller ground to the Arduino ground to make the logic levels actually communicate between the two devices.

You likely also need some 5V->3V level shifting to isolate the Arduino's output levels from the controller's inputs.   Top of the line would be MOSFETS, but you could possibly use a voltage dividing resistor pair.  See the hints in https://arduino.stackexchange.com/a/419/6628