I have not seen any results from the Tribunal in League of Legends yet. I have been going on it every day, or every other day, and so far, I have yet to see a situation where I was on the fence about someone.
Let me elaborate, when you look at a single game, you have a tendency to be more forgiving, but usually, by the time they are at the Tribunal, they have 8-12 games worth of complaints from several people each game, you start getting a good picture of what is happening with this player.
Still, since they force you to wait before passing judgement, it forces you to at least take a peek at what is happening. I do look at the games, to see if their behavior is consistent within each. In the end, I do it because I want the game's experience to be improved.
If you've had any experience with the Tribunal, let me know in the comments.