I haven't actually played URF. I think it's kind of dumb and I'd rather play game modes that actually improve my skill. All of my games since it launched have been as ADC or jungle in plain old boring SR. However, URF is a pretty interesting thought experiment. What's broken and why? What isn't broken?
First, I want to look at revive passives like Anivia and Aatrox. Are they affected by URF? I don't play either character so I don't know. I know Revive was removed from URF, lol. If Aatrox passive is affected by URF then he's really good.
"Any character with a dash is good." Pretty much true, especially if ranged. Although Ezreal is the biggest stand-out, I want to point out characters like Graves and Aatrox too. Sejuani's dash is probably ok though her kit is probably not as good.
Chain CC is a really big possibility. Most of us know about Alistar but there's also Ahri, Poppy, Nami, Leona, Brand, Blitzcrank etc etc. LeBlanc is the worst. Perma snare and perma silence all in the same go, on a character with an escape. Perma ban, in my opinion. Any kind of chain CC is just nasty.
Characters with burst resistance are especially good. Blitzcrank passive is really good if it is affected by the 80% CDR. Poppy's passive is also crazy; it takes a lot of hits to bring her down, even in URF. Characters with inherent tank seem very strong. Singed, with his perma ultimate, is probably a lot more dangerous than he appears even if he doesn't get a lot from 80% CDR otherwise. Same with Leona and her perma Eclipse.
Characters that can unload lots of damage, especially from range, are great. Jayce and Ezreal are among the best. I think Nidalee is highly overrated; her spears are rapid but everyone else gets more rapid stuff too and Nidalee can't deal with getting jumped on. Ziggs is the real terror. Multiple minefields AND ranged poke AND a near-global ult AND a dash (sort of?) What the hell.
Initiation is really crazy with Flash being up every minute. Galio is probably the biggest. Morgana and Kennen are probably OK. Hard to say with other characters. The best initiations are probably actually picking tools like Nautilus' ult/hook, Blitz hook, Morgana binding, etc.
I'd say that you need either insanely good one thing (Galio ultimate) or really good in lots of things (Ezreal). Melee characters are a big liability. I think you basically need to be as good as Poppy (nigh-invincible and chain CC and huge damage) or be not really a melee (like Maokai or Gragas) to be good in URF. Due to the huge amount of gold and infinite mana, Poppy is probably decent. Akali is probably also good due to chain Shroud (takes away from the number of trinkets enemy team can have, since they need to Sweep her) and huge huge damage.
Jayce, Ziggs, Karthus, LB, Alistar are probably must-picks if open.