The remote for my Hampton Bay Redington III ceiling fan no longer works even though the LED lights when a button is pressed. Does this mean the receiver is bad?

Receiver bad or the transmiter. You replace them as a set so it really doesn't matter much.

B, you only replace them as a set if the fan has pullchains.

It is USUALLY the receiver that is the problem.

Ceiling fan remote controls - help and FAQ- Ceiling Fans N More