The answer to this question concerns my favourite character in this play: Feste. As always with Shakespearian fools, they often, under the guise of their foolishness, show far more wisdom and intelligence than the other "wiser" characters they are there to supposedly "amuse." A key speech to consider in the character of Feste is Viola's soliloquy about wisdom and folly that she delivers after talking with Feste in Act III scene 1, when she is disguised as Cesario:
This fellow is wise enough to play the fool,
And to do that well craves a kind of wit.
He must observe their mood on whome he jests,
The quality of persons, and the time,
And, like the haggard, check at every feather
That comes before his eye. This is a practice
As full of labour as a wise man's art;
For folly that he wisely shows is fit,
But wise men, folly-fall'n, quite taint their wit.
Note how this is actually a rumination on the role of the fool and how intelligent and wise fools need to be to play their role successfully. Many productions play Feste as a character that actually is able to work out what is going on. It is strongly suggested in places that he sees through the disguise of Cesario, and the fact that he travels between the two households, performing for both Olivia and Orsino, suggests he is able to take a global view that other characters lack, therefore giving him knowledge about what is going on in Illyria. Therefore, for all of these reasons, Feste is most definitely the wisest character in the play.