What is the maximum Elo rating that Magnus Carlsen could have if he loses the 2014 WCC to Anand? What is the maximum Elo rating that Magnus Carlsen could have if he loses the World Chess Championship 2014 to Anand? Would this result from a 6.5-5.5 result or would the relative rating change mean that if he lost more quickly that would be better? Also, are the players playing 12 rounds regardless of the current score?
In a hypothetical situation where Carlsen knew that he was going to lose the match and his number one priority was to preserve his rating points, then his best option would be to lose as quickly as possible. As pointed out in this question (EDIT: Values updated as per JiK's comment below), Carlsen will lose 1 rating points for each game he draws with Anand. The only way that he could extend the match and not lose points in the process would be to win as many games as possible before losing. 

Assuming Anand wins with the minimum possible score of 6.5/12, Carlsen will lose a minimum of 17 points (11 draws x 1, 1 loss x 6). This does not change if draws are replaced by a win for either side.

However, if Anand was going to win with a draw and 6 wins, then Carlsen stands to lose
 37 points. If Carlsen somehow knew this would be how Anand would win, then he could attempt to interpose as many of his own wins (up to 5) as possible, ameliorating the loss somewhat. Even in this situation, Carlsen's best result remains a net loss of 17 points.

EDIT: And to answer the question as it is worded in the title: Carlsen's rating (assuming no results outside the WCC match are counted) would be no more than 2863.0 - 17 = 2846.0