In OTB tournaments you are not allowed to ask your opponent to resign and here I consider it very bad manners.
However, on the other hand players who play on in a hopelessly lost position can also annoy me at times. In OTB it is also considered a breach of etiquette to play on in a hopelessly lost position - although it is of course legal to play on.
I understand the point that there can be an educational gain by those players to see the win - i.e. therefore they play on to learn something. Fair enough.
I have only once asked a person here to resign. From his rating and being a queen down there was no point in playing on - with his rating he knew for sure that such a position is beyond salvation and with 3 days per move he knew that a player with my rating does not blunder into a stalemate. Furtheremore, this player had explicitly stated in his profile that he himself hated people who did not resign in lost positions. It seemed he did not follow his own words in reality...but then again the longest distance in the universe is the distance from words to actions...:-)
If a player has a high rating then they get no educational value of playing totally lost positions I think (I have around 2000 FIDE ELO for those who wonder).
Sometimes it is not obvious if a position is lost - but in those in extreme cases of being a queen or rook down with no compensation a strong player should always resign I think. He cannot give the excuse that he did not know such a position is lost - a complete beginner can use this as a legitimate excuse but not a strong player.
Personally, I wouldn't ask my opponent to resign. That seems rather "un-gentlemanly" for something that is supposed to be a "gentleman's game", but that's just my opinion (the exception being: something like announcing "mate in 5" when there is a forced mating sequence).
Nor am I suggesting that a player should resign once such-and-such position is reached (e.g., one person is up a piece, say). The rules of chess state that a player can win either of three ways: 1. checkmate the opposing king; 2. opponent resigns; 3. opponent runs out of time. A win is a win - it doesn't matter which of the three victory conditions you achieve. If the opponent doesn't want to resign, let him play on. If your position is "won" then finish it off, show him the checkmate.
I also don't think it's right for people to get upset when their opponent refuses to resign - in many cases they may simply not understand why their position is lost. And I think it's even worse for people to get upset when their opponent does resign - you just won the game, what are you complaining about?!
Personally I would resign once I understand that my position is hopeless, but play on if I feel I have a chance to salvage a draw (or steal a win!). It seems that many share my view on this. From the comments in this thread it would appear that most people take a common sense approach to this issue, and the best advice would be to ignore the sore winners who complain about opponents not resigning (or resigning!).