The straightest hack would be to modify the remote control by carefully removing the keypad and identifying the track matrix which exists behind it. Imagine the keypad as a simple 6 x 4 matrix and the controller chip alternately applies 5v to each of the 6 rows in turn, and when you press a button you bridge the crossing point between one row and one column, and at that point in time the controller looks at the 4 column lines to see which one is at 5V. These are only sketchy details but a realistic and common way to operate a keyboard....imagine it scanning across the rows 10 times per second. What you would need to do is to detect when the 5v of the row you're wanting to set active is switched on and have the Pi send 5V (or 3.3V ) down the column line you want to emulate. Crude, a bit fiddly, but doable.
A cooler thing to do is to pop an IR receiver on the Pi and 'record' the output from the IR remote when each button is pushed. Then use a suitable IR LED to 'play back' the commands you wish.
There are in fact well known IR remote control control protocols so if you're lucky you might find a part number on an IR control chip in the controller, the datasheet of which will tell you the protocol. At the worst you will see an epoxy blob on a pcb with no markings and you're back to recording the IR signal at up to 60kHz frequency, and possibly much less. I would probably aim to use a PWM output at 38kHz, 50% duty
Here's an example of someone doing something similar