I read the story sometime around 1993, and I seem to remember it being part of some anthology.
The POV character is immortal, in that whenever he is killed, he regenerates. While he is whole, he doesn't feel anything--taste, pain, etc. Immediately after he is renewed, he can feel again, but sensation degrades quickly.
As an immortal, he "performs" for individuals in high-society, being killed repeatedly for the amusement of others. I seem to recall the story taking place immediately after (or during) one such performance.
He's not the only immortal. He was part of some sort of voyage, and his crew-mates were likewise cursed.