How do I operate a tuner of digital satellite receiver using a microcontroller?
I would get an extra remote control for the receiver, and mount it in such a way that it sits on top of receiver, and shines down on the IR window of the receiver (perhaps using a mirror).

Then hack into the remote control, and wire into the buttons that you need to operate the receiver.

The easiest way to do this, without trying to figure out the input circuit of the remote, is to wire reed relays across the contacts (note that the buttons probably don't actually have contacts in themselves, but may operate dome switches or some other type of switch contact located on the PCB of the remote controller).

If you want to get fancier, and use less relays, you will probably find that the conact switches are arranged in some sort of array (for example, 6x6 would use 12 wires, but would allow for 36 buttons).  So then you just have to control the appropriate two wires for each button.

In either case, I would attach a couple of wires to a single button first (such as Mute), and try connecting them manually and observing the result.

Beside reed relays, it would probably be possible to go all solid state using FETs, but you would need to reverse engineer more of the circuit.