Why do you press “f” to pay respects instead of “r” for respect?

Because it originates from a video game. Specifically a Call of Duty game. Video games do not choose the letters you press based on the start of the word they're associated with. They pick the letter based on what is most comfortable for the user to press. WASD is used for movement in most games on PC, and F is just a really convenient button to use as an action key because it's really easy to press with your index finger while your hands are on the WASD keys.

Because 'r' is reload.

The PC version of the game "Call of Duty: Advanced Warfare" made players press F to pay respects. That became a meme, and people have been pressing F ever since.

Why the video game made players press F, I don't know. Maybe the R key already did something (possibly reload).

edit: extreme weeb also made a good point.