I also am curious about a particular issue. I am 100% so far on agreeing with the majority since I entered the Tribunal yesterday, but a couple of the 20 cases I tackled today were kind of tricky to make. What do you look for if the reason was "Intentional Feeding"? This is not really something that is obvious unless there is some discussion in the chat logs about it. I personally look first at the number of deaths the reported user got, then usually checking the logs to see if either the allied team or the enemy team talked about such and such champion being easy to kill or feeding. This kind of issue would be easier to judge if we had gameplay footage, but I am not suggesting we have anything like that, just trying to say how, imo, that issue can be difficult to find.
Alas, I was hoping to find cases I could pardon, but if I find any evidence that, to me, makes me believe beyond any reasonable doubt that the person in question did break the Summoner's Code, I have no choice but to choose punish. I hope this won't be the case 100% of the time, but I am to judge based on what logs I can see, though. All I will say is that I will try to judge as fairly as I can.
You can also check the chatlog for teammate reactions, and usually that gives you some clue to what's going on.
It'd be nice if they added the killfeed to Tribunal reports though.