A Youtube video goes through the pronunciations of all the names in Twelfth Night. It can be found at the link provided below. As you can hear, the name is pronounced Fest-ee, with a long "e" sound as the second syllable.

Feste is most often referred to as the "Fool" in the play. The one time he is mentioned by name is in Act II, scene 4, when Curio identifies him to Orsinio, saying:

Feste, the jester, my lord; a fool that the lady Olivia's father took much delight in. 

Feste's being primarily designated the Fool, a generic label, emphasizes that he has a dual function in the play. He is not just a character but a voice that speaks the truth about what is going on with the other characters. He is thus both inside the play as Feste and yet stands outside of the action as a commenter on what the others do.