What exactly is this "heating unit" - is it something portable, or is it mounted on a wall or something? Is it electric?
If it is portable and electric - and if you are comfortable and understand mains AC - then likely that knob just controls a simple bimetal thermostatic switch. It probably has only two terminals - one terminal connected to the heating coil (with the other end of the heating coil connected to the neutral side of the AC mains), and the other terminal connected to the "hot" side of the AC mains (there may also be a switch and/or a motor for a fan involved, but likely all of those are in series with each other - though a motor/fan might be in parallel, and/or have a switch to control it).
So - if you wired a simple SPST relay (or a solid state relay) in parallel with the thermostat - then controlled the relay with the Arduino - you could dispense with trying to control the thing with a mechanical solution.
This heating unit is something mounted to the wall which goes directly to the vents, so unfortunately this solution will not work.
Maybe you could use a servo to turn the knob. They come in a wide range of sizes and are very easy to control with an Arduino.
Servos have all their electronics inside so you don't need a shield. But the servo does need its own power supply - don't try to power it from the Arduino 5v pin.
This sounds like a great idea! The servo does not need to turn more than 180 degrees to fully open or close the dial. Do you think you could link me to a servo that provides enough power? Every time I've looked up a servo I only find ones that power themselves off of the Arduino itself. Or at least give me a keyword to search for a motor with an external power supply :)! Also, do I need external power regulators and switching regulators?