How do the other characters in the Twelfth Night view Feste?

Feste is a professional fool, or jester. This places him in a position where he is permanently isolated, estranged from so-called polite society. As a fool, he's been granted the fool's pardon, which allows him to say things about other characters that he wouldn't otherwise be allowed to express. He's also a highly intelligent man; indeed, he's arguably the most intelligent character in the play. His acute intelligence gives him a great insight into the myriad flaws and foibles of everyone else, an insight which he then passes on to us, the audience.
Despite this, Feste isn't really taken all that seriously by anyone else in Twelfth Night. At best, he's a source of entertainment and amusement; after all, that is his job. But the prevailing social hierarchy is so rigid that no one can really look upon at Feste as an equal, cannot really accept him as a fellow human being in the fullest sense of the word. For all his sparkling wit, inventive wordplay, and winning personality, he is still a servant, and as a servant must work hard for his keep.
Feste finds himself torn between the dictates of a free-spirited soul and the sometimes onerous responsibilities of his role as court jester. Every so often, Feste feels the overriding need to take off, to break out from his socially-imposed role, even if it means running the serious risk of being punished by his mistress Olivia.
It's tough being a fool, especially when no one takes you seriously as a person. It's even worse when your jesting skills appear to be slipping, as Malvolio indicates:

I marvel your ladyship takes delight in such a barren rascal. I saw him put down the other day with an ordinary fool that has no more brain than a stone. Look you now, he’s out of his guard already. Unless you laugh and minister occasion to him, he is gagg’d. (Act I Scene V)

Once a fool stops being funny, there's no further use for him. Feste's isolation in the rarefied world of Twelfth Night is confirmed at the play's ending when he's left all alone. He doesn't belong with Olivia and her lovers, but nor does he belong with Sir Toby and Maria, Sir Andrew or Malvolio. He is an entirely separate entity, complete in and of himself, but destined to be forever defined by the onerous, wearisome burden placed upon his shoulders by a society that steadfastly refuses to acknowledge his individuality.