Yep I'm gonna keep updating this cause why not, new entries in italics.
Garbage Tier
Singed
Yes, I know that permanent R mode sounds cool and all. But his primary source of damage, poison, does not increase at all in this game mode. But the biggest offender is that Singed can't outrun anyone anymore. A champ with any dash is now speedy gonzales (an overrated trait in URF I believe, but it makes Singed shit in comparison).
Low Tier
Lux
Here again the shining light is the ultimate, a now beautiful 10 seconds. Unfortunately, Lux otherwise gains very little from this mode. Lux is a calm, tactical character. She just can't function under this level of crazy, and forget using her passive outside of a duel.
Poppy
She just felt really slow at dealing damage unfortunately. I could swear her Q used to do more damage, but Poppy is def more about the chase and even when you can stun she just doesn't offer enough damage.
Vayne
The first big blow right as you start is learning that you cannot spam Q. In this mode you also just feel slow, even with R up. Just an awkward champ uniquely hurt by URF mechanics.
Mid Tier
LeBlanc
Like I said, she is overrated. Yes, I've actually moved her down from high tier. There are just many things that make her awkward. The most of which being how the cooldown on your W will get lower than its own subcast. LeBlanc can solo very well early, and if you get fed your team has a strong chance of winning. But I feel there are better champs who can win for their team even without lots of early kills. After playing many better champions, and consistently stomping opposing LeBlancs, I cannot justify putting her in a higher tier.
Leona
This was the first character I played, so I couldn't know going in what her shortcoming would be. Every character has 25% tenacity, so not only is she not a perfect lockdown, she can't even combo her R with Q well anymore, especially when you consider all the people who are going to be blinking out of every lock. Stil, she has perma-W and offers about as much to a team fight as always, it's just not a tank friendly meta.
Nasus
Farm on overdrive. Sadly, the value of wither is diminished so Nasus has a harder time than ever catching people. Now if you decide to backdoor though, you will knock over towers in seconds.
High Tier
Ashe
I feel like ranged AD carries are super underrated in this mode. Zhonya's is the most common armor you are likely to see. Ashe also brings to the table one of the most potent pokes in the game, making laning a breeze. Having the ult more often also lets you pull off some wild stunts. If you don't manage an R+teleport combo at least once per game, you know you're doing it wrong.
Hecarim
I hear some people would say this character would be higher, and I think he's really damn good but there's surprisingly more broken characters. Not much needs to be said about Hecacopter, he is a rare good MS character in that he can Perma-E and it stacks. Any character that already had a low cooldown ability is bound to be good, and Hec does not disappoint.
Heimerdinger
This one hinges on God Tier folks. Forget your fucking blinks. You really think Heimer is unsafe without a mobility option? Well prepare to eat 3 turrets then. He will never not have three turrets up. Even if you kill him, he'll probably kill you right back. And that goes for 1v2s as well. Spamming his W and E are also wonderful, and R works great with all abilities. The only thing keeping Heimer from God Tier is a relatively hard laning phase.
Miss Fortune
You thought she dumped on lanes before? Meet permanent W. It takes very rare combinations of opponents to not be able to win lane. In a squishy game such as this, her ult is also sick.
Graves
I feel like Graves is everything that people think Ezreal is right now. Sure his dash isn't as good, but his Q is much more suited to this kind of game mode and his R is pretty sweet too. Between a 3 second dash, a redonc poke, and all the buffs that ranged carries got, Graves is very much a character that win with any amount of early kills.
Rammus
This was a pleasant surprise. Rammus always dealt a fair amount of damage for a full tank, but in this mode he just chews through squishies before they can do much about it. He can also just jump around the map straight up icing people, first as ganks, and then even without. Then in team fights that taunt is a major asset.
Sion
As usual, people are going to come to Sion for one thing and it's going to backfire. That would be his shield. Even though the cooldown is 2 seconds, you still have to wait all 4 seconds to blow it up. That feels like an eternity in this mode. So why the high tier? Simply because of his Q. Not only is it one of the premier lockdown tools of this mode, it also hits like a ton bricks. And even though you can't rely on your shield exploding, you can put it up almost immediately after an opponent breaks it. Even his E and R have late game significance in a mode with a ton of backdooring, not to mention that his R is up permanently.
Taric
This is the real surprise sleeper. Everyone is busy banning Ali, Nida, and Sora, but nobody picks or bans Taric. Which is good for me I guess, because he has the truest permastun in the game, and the biggest heals. You can even use his ult twice per fight sometimes. How is someone this crazy not God Tier? Well... he's oddly horrible at even 1v2s. His W almost feels like a detriment to use, taking precious time you could spend healing or stunning.
God Tier
Alistar
Soraka can kill 1v5, but Alistar can tank 1v5. Any heal is abusable, but an AOE heal with an AOE stun? You can knock people around like it's nothing now, that ult is up all the time, and even his passive is deceptively amazingly at level 1.
Sona
Allow me to play you the song of my people... QWQEWQEQWEQWEQWEQWEWQEQWEQWEQWEWQEWQEWQEQEWQEQWEQWEQWEQWEWEQWEQWEWQEQWEQWEWQEWEQWEWQEWEQWEQWEQWEWEWQEWEWQEWQEWQEQWEQWEQWEQWEQWEQWEQWEQWEQWEQWEWEWQEWQEWEQEWQE
RIP in peace
Soraka
Just... tear to my eyes beautiful. As a longtime (and sometime offensive) Soraka player, going 30/5 was just glorious. Her Q is already way stronger than Hecarims, but when you add in constant heals and the ability to neuter other purported "god tier" characters (here's looking at you LeBlanc, Urgot), Soraka may be the best this game mode has to offer.
Overall Thoughts
I guess I just don't think URF mode is as unbalanced as people say. There are certainly outliers, but they fall well within the 6 bans. In a game mode with 5x gold, the only way you can really get shut out is by getting behind in levels (in rare games the opponent can get up to double your level). Otherwise, every champ should be operating at high capacity by 20 minutes even if you're 3/10.