These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Feste gets a lot of this. His generally witty lines leave the majority of viewers genuinely amused and enjoying him as comic relief, but his somewhat bittersweet closing song ("the rain, it raineth every day" - see The Cover Changes The Meaning, on the main page) make others view him as a subtly tragic Sad Clown. Finally, the ludicrous extent of his Disproportionate Retribution (again, see the main page) makes others argue he is a Monster Clown with total sociopathy, willing to unmake any character as thoroughly as he did poor Malvolio if given half a chance to get away with it.
Directing a production of Twelfth Night takes on an extra dimension if you're familiar with Alan Gordon's excellent Fools' Guild mysteries, to whit: Feste is a secret agent, engineered the twins' "shipwreck" in order to stabalise the political situation in Illyria, and falls in love with Viola to the extent that after his assignment's success, he spends a lot of the rest of his life trying to drink away the heartache. Oh, and Malvolio is apparently working for Saladin. Introducing these concepts to the actors playing these roles doesn't remotely translate to the audience, but is a hell of a lot of fun.
The respective sexual identities of Orsino, Olivia, and Viola are very up in the air, considering that Orsino falls in love with his page "boy" and Olivia falls in love with a woman that she thinks is a eunuch. And unlike Rosalind in As You Like It, once outed Viola is never seen again on-stage in her woman's weeds (although modern productions may have her come out in a dress for the curtain call).
Malvolio could be cast or played as being Olivia's own age, or an older but still handsome fellow, or as a man way past his prime but still daydreaming that the Countess will fall for him.
Some productions imply that Feste knows Viola is actually a girl, and uses lines such as - "Now Jove, in his next commodity of hair, send thee a beard" - in order to tease and bait her. (They did this in the 1996 film, although Feste had an unfair advantage there, since he saw the survivors of the shipwreck getting washed up on the shore.)
Esoteric Happy Ending: More cynical eyes can view the ending as this. Yes, Viola and Orsino are (probably) happy, all well and good, but what about everyone else? Consider:
Olivia's married to Sebastian simply because she mistook him for his cross-dressing sister, who's the one she actually fell in love with,
Sir Toby has a serious head wound and might well be cast out by Olivia for everything that he's done to Malvolio; same for Maria on that last part,
Sir Andrew's been violently rejected by someone he honestly believed was his friend and trying to help him,
We're never told what's going to happen to Antonio, who only got caught because he loved Sebastian so much (in the 1996 version Orsino frees him, but he goes away on his own, bereft)
Do we really need to elaborate on Malvolio?
And Feste even sings a melancholy song about the rain raining every day. Hardly cheerful stuff.
Ho Yay: Some more Freudian interpretations of the play suggest it's actually intended to be a major theme. It's particularly blazing between Antonio and Sebastian.
Jerkass Woobie: Malvolio, hands down. Even the characters admit they went too far in humiliating him.
Narm: Critics have pointed out that in Viola and Sebastian's reunion scene, the whole dialogue of "My father had a mole upon his brow" feels really artificial and disrupts what is otherwise a very honest scene.