Guessing what may be wrong, a separate interface relay will solve the problem, but how do you connect it? This could be a transistor, FET a solid state relay or a conventional relay, though some of these may not work so readily in all cases. The relay is the most general solution. I don't know what level you are at, so I am describing basics here.
You need a multimeter... Buy a cheap one if you don't have one.
You will need to learn...
How to use the ohms range for measuring continuity:
This works with just the test leads, when switched to the ohms function. When they are touching it should be almost zero ohms, and when not touching "infinity", open circuit or a full scale reading. The range can by ohms X 1 or ohms x 10. Don't connect the meter to a voltage source when in the ohms range!
How to measure volts like 24V:
Switch to the DC volts function and connect the test leads across the supply (pins 1 and 3 you mention). The range needs to be greater than the voltage, perhaps the 200V range.
Determine what the output of the remote control does first. It may be 24V between the pins 1 and 2 you mention or 0V, representing on or off. Try that first. If there is 24V that switches on and off, you have a voltage output. If it is not this way, it seems like the output is a relay contact closure (both states are zero volts). Using the ohms scale you expect to see a short circuit in one state and an open circuit in the other state. You have a ground switched output.
Now you know it is switched 24V or switched ground.
Determine what the switch on your circuit board does. With power off, and not connected to the remote control, use the ohms range to determine whether the switch is closed or open when the switch is active. Closed will be zero ohms, open something greater. Check if one side is connected to ground of the circuit board (probably the negative side of its power supply or supply (the positive side of the supply).
Using a relay with a 24V coil and one changeover contact to explain.
If the remote switches 24V to the output, simply connect the relay coil across the output pins 1 and 2. You should hear it operate and release as the remote is switched, and see its contact changes using the ohms range on the multimeter. If you have only a contact closure in the remote, power the relay from 24V and use the contact closure to complete the coil circuit, so the coil is connected between pin 3 and 2.
Now use the contacts of the interface relay to make the same connection as the switch on the PCB does. If this doesn't work, maybe ask a new question with the new information supplied.
If you use a transistor, solid state relay, etc there are further considerations. This would require more analysis, and so I recommend you go with the relay. Maybe ask a new question with the new information. It is quite likely the solid state relay won't work.
N.B. try to get a relay with gold-plated contacts.
One other idea is that if the control unit switches 24V (probably not good for your circuit board) you could rewire the relay contact in the controller so it switches ground instead (pin 1 instead of pin 3). That may suit the switch on the circuit board, assuming is also switches ground. You could also extend both the relay contacts out (so there are 2 isolated relay contact connections) and connect them across the switch on the circuit board.
This should get you started in the right direction, and help you solve your problem - the interface relay may not be essential, but we are looking for a straight forward solution here in the face of limited information.