Suppose you're the PC chair, and someone who has submitted a paper asks you NOT to have the paper reviewed by a specific person on the PC. Do you honor that request?It's an interesting question -- that I hope others will comment on -- though as a default my answer would be yes. I certainly have had run-ins of sufficient severity with various people through the years that I would not want (and would likely ask) for them not to review my papers if the issue came up. Looking at it from the other end, if I am on a PC and those people submit a paper, I make sure not to review them. (Usually it is sufficient simply to rank them low on my list of desired papers, but I have also told PC chairs in advance I would not review certain papers if they seemed likely to head my way.) It is not that I actually think I couldn't give a fair review; it's that I think it's inappropriate, in such a situation, for me to give a review in the first place. If as a PC member I have the right (actually, I would say, a responsibility) to refuse to review a paper under such circumstances, it seems fair that a submitter can ask for a specific PC member to not review a paper as well.
Context does matter, though. In the networking conferences I have served on, this is standard -- PC members and submitters are expected to list their conflicts. Indeed, one issue that seems to have arisen lately is that there is suspicion that some people submitting papers are abusing this right, listing people as conflicts when they are not because they are known to be "challenging" reviewers. While I'm skeptical this sort of gamesmanship gains anything (challenging reviewers are usually calibrated appropriately at the PC meeting), it is a concern that once you open the door to such requests, you may need to make sure the privilege isn't abused.
For theory conferences, where many people seem painfully unclear on what "conflict of interest" even means, I'd grant such a request as a matter of course.