When an answer becomes down-voted sufficiently to cause it to have a negative score, is it good practice for the author to remove the answer? The reason I ask is that I recently posted an answer to the Dining-table question, which has been down-voted, and I am wondering whether this means I should remove it.
You are under absolutely no obligation to delete answers with downvotes, unless it becomes clear that the answer doesn't in any way answer the question being posed, and even then it's your decision.
People should not do drive-by downvoting IMO, without thinking about how to improve the answr (this goes doubly so for downvoting questions).
If I start to believe that my answer to a question is irrelevant or not good enough, I would either improve it or remove it. Down votes can be a sign, but the decision shouldn't be based just on down votes, but rather your opinion about the relevance and quality of your answer. Probably one should do the same thing even for answers with net positive votes.
The SE system does currently encourage posts with scores of -3 or worse to be deleted, by awarding badges for their deletion. But a few down votes don't necessarily mean much.
Questions rated -5 or worse seem to be pruned from the lists unless one goes in search of them. Orchestrated downvotes could therefore become a denial-of-attention tactic. (I'm not sure what happens to answers rated -5 or worse.)
I agree with the other answers about trying to improve a post with negative scores, though sometimes a post just isn't going to find any love. In general, a post that has had 20 +1 votes and 20 -1 votes is likely to be at least interesting, but one which has not had any votes at all is probably not right for the site for some reason. Maybe a future version of the system will have different metrics...