I'm against Role-Players making and running characters that are designed to have goals that conflict with the ordinary goals of the party. This includes "moles" and "possessed" characters. My main concern is that it's done too often; I'd be more surprised if some players had a character without a complicated issue that never really comes up in the game but prevents them from really helping.
It makes "Character information" and "Player information" much more difficult to segregate, in most situations, a player that was concerned about it could just ask for clarification: "Did you say that in character?", but a PC with a secret agenda can't ask that because it would be very suspicious. The information problem runs the other way too, the secret actions will require additional adjudication by the GM, which will almost inevitably be noticed by the other PCs, unless they are completely oblivious. (This is why I have never detected a hostile PC ever)
There are a lot of things that go unspoken in an RPG, a player doesn't say that he turns on light switches, or goes to bed, or whether he locks each door he closes, or whether he checks his watches, or does regular virus checks on his computer, etc. But each of these actions, and endless more, could reveal or betray a PC that's acting against the interests of the party. The GM is forced to RP the PCs on behalf of the antagonistic PC, who ends up essentially acting against brain-dead players, no GM can imagine all the nonsensical things a party of PCs will do, and he has an obvious incentive to not catch the secret PC in action: that would end the entire plot line, and could easily lead to the death of a PC, or even the entire campaign. Thus, the players feel cheated that their "implied" actions didn't actually happen. The GM can't ask what sort of precautions are being taken because, again, this creates obvious problems with information between player and character.
Note, I don't mind if a character develops in such a way that they must oppose the actions of the party, I'm against a character designed with that in mind.