Do summoners get any form of reward if they get cases correct on the tribunal?

Nope

So far, no.

But they have hinted at rewards being implemented some day. But if you're expecting something like IP, RP or skins I'd say chances of that are close to 0.

Maybe some sort of social reward like forum avatars, forum icons or summoner icons.

They get a shiny number. That number goes up when they're right.

As for tangible rewards like IP or RP, nope.

Dang... well thanks for posting