Well, to start with, I find him annoying. He's good for a couple cheap laughs, I guess, but that's not really a good thing. He contrasts with the tone of pretty much the rest of the movie.
Next, let's consider just how
creepy his desire to see summer is. It makes
this song seem a lot like a song about a kid wanting to, say, play in an incinerator while constantly making reference to how he'll burn to death. Seriously, what the fuck Disney?
Moreover, he doesn't serve any purpose. What does he do? He's created by the two little princesses then re-created in Let It Go, then he comes to life, then he shows up and makes his cheap laughs. Through the movie, aside from characters talking to him and once throwing him, he doesn't really do much. Then he, spoiler alert, finds Anna after Hans has left her to freeze--but anyone in the castle could do that! A servant, or another visiting noble, or heck, the Duke (give him a shot at redemption and/or character development). He gives some advice that convinces Anna to try and go to Cristof...which, again, most of them could have done somehow or another. Finally, he almost melts, and Elsa has to save him. He isn't even needed as comedic relief; if you think the movie is too short on humor, add more Anna antics, or Sven talking with Cristof, or the Duke, or the trolls, or find a way to make the more serious characters the butt of a joke.
Let's compare him to a character I've realized he's a lot like*: Sid from Ice Age. Both are dumb, but not quite as dumb as they look (pretty close, though); both are the most caring characters in the main cast; and both are largely comedic relief (among other things). Now let's look at the differences. Olaf is stupider than Sid by a wide margin, and his humor tends to be more on the "stupid slapstick" side of things. Olaf's stupidity is largely ignored or adored; Manny and Diego get annoyed at Sid, so it's less annoying to watch his annoying moments. Sid's humor fits in with the lighter tone of Ice Age better than Olaf's fits with Frozen, so there's no dissonance. Sid has multiple dimensions (well, a couple); Olaf's basically got one, maybe two if you count his unexpected proficiency at "love expert"-ness. Perhaps most importantly, Sid's was actually needed. The herd wouldn't have stayed together if it wasn't for him--it wouldn't have even formed! On the other hand, Anna is driven by her love of Elsa, Cristof by obligation towards Anna (and the promise of a new sled, and growing affection), and Sven because of Cristof. You don't need a snowman to keep them together. Moreover, Sid does stuff, stuff which no one else could do (e.g, the melon, the fire, the distraction); you feel like he's an integral part of the story, rather than an add-on. And while there are humorous characters other than Sid, aside from Scrat's occasional appearances none show up for more than a scene or two.
*Actually, I've noticed that there are a lot of similarities between Ice Age and Frozen--not enough to think that the latter was based on the former or anything, but enough to be notable. But that's for another time.
