What happens when I get champion essence, but own every champion?

Not much. On the PBE there was nothing to do with those champion shards. The only use I can think of is saving them so that you can instantly get new champions for free when they come out.

I think you an disenchantbthem for more essence

There was some talk about it here. Can't say if it was ever implemented.

http://boards.pbe.leagueoflegends.com/en/c/general-pbe-feedback/XT7eu7jz-champion-mastery-level-6-7