single word requests - What would you call a person who is being replaced by another

Predecessor

A predecessor is exactly what you describe. A person whom was replaced by another.

Barack Obama's predecessor is George Bush Jr.

This word isn't exactly thrown around in everyday speech either. It's usually reserved for higher up positions (like my presidential example). That doesn't mean it couldn't be used for other jobs though.

Jim Jones is the predecessor to the fry cook at Wendy's

meaning Jim Jones was the fry cook and has since been replaced.

The person who replaced Jim Jones is called the successor, more particularly Jim Jones' successor.