One of the biggest problems in any game I've run that deals with "Alignment" are those roleplaying choices that the players make that technically violate their listed alignment. It's very difficult to judge what would just let slide or when I, as the game master, should call them out on it, or at least question their judgment (are you sure your lawful good paladin would be ok with slaughtering all of these orcs in their sleep?). But as I develop my "custom" game world, I'm going to be increasingly insistent that actions have consequences in terms of alignment. It works both ways, too. If a PC can slowly become neutral (or even evil), an evil NPC could also become more neutral (or even good).
I do believe that these sorts of alignment change questions, when relevant to the PC, should be discussed with the player outside of game, and perhaps privately. Other characters might not notice the change at first, but become slowly aware of changes in the character's personality that do not mesh with what they usually expected from them. In the case of a divine character, I don't see why they should lose their divine powers (even in the case of one who worships a lawful good deity), but the powers become warped somehow. Or perhaps they change allegiances. As they stray further and further from the path proscribed for them by their deity, their holy symbol darkens, tarnishes, fades, or chips, something which can be noticed by their fellow adventurers. This can be played out in game, as well. Perhaps a crusader for Bahamut begins to feel disillusioned by being constrained to only do good, even when it might, ultimately, allow an even greater evil to flourish, and instead strays to the side of the Raven Queen as the impartial Ultimate Mediator.
Corollary to this idea of alignment change is the idea of betrayal. I have never actually had a game where one of the PCs betrays the others, but there are some games that make it much easier (such as Call of Cthulhu). Chief in my mind here are the Horus Heresy Warhammer 40,000 novels where the titular character, Horus, is corrupted by chaos, and it slowly becomes more and more apparent to Garviel Loken as time goes on, as he acts further out of character and quickly begins to make Loken legitimately concerned. Certainly a PC betrayal would, in most situations, result in the PC being ejected from the party and, at best, becoming an NPC. But is there a situation where a character could become evil in a group of good PCs and still remain mostly in line with the party goals? I can only jokingly think of the Order of the Stick, where the other characters have to keep Belkar in line, because he is Chaotic Evil, something which they have to keep a secret from NPC paladins who might otherwise want to smite him.
An evil (but otherwise harmless) PC might create similar hijinx when dealing with an uncertain populace, and constantly having to corral that character in, given an accepting group, might create fun and funny roleplay experiences. I will be very interested to see what Rob Schwalb does with 4e Book of Vile Darkness, because I am constantly trying to think of ways to create more situations than just "good characters kill evil badguys, everybody win." Life is more interesting than that, so why shouldn't the game be more interesting than that? Supposedly good people are compelled into doing evil things all the time, whether by circumstance, trickery, or plain folly. A game which allows for the "banality of evil" would probably be more interesting, but I've just never had the opportunity to put something like that together.