I have it done in my machine, but am unable to share any of the code as it was pieced together from various sources, a few of which I was told not to share. - However the basis of, as least mine, was incredibly simple.
Press a button on the screen, which sends out a signal from the arduino to the actual solder point of my HVAC buttons to simulate a button press. The buttons on screen don't know what they are actually doing, and are just sending out signals. I'm not a programmer myself and found someone to help me, without any experience with the arduino, did it it only a few hours.
If you have any other questions I'll try and help as much as I can.
Here is an image for reference.