As far as i know most MMOs don't run heavy anti cheat stuff as they are built to not trust the clients so i suspect a lot will work fine.
Some i know work perfectly fine are:
Guild wars 2, both through lutris and proton. Back when i originally swapped to linux i had some issues getting the commonly used dps meter (arcdps) to work with proton. It requires some optional features to be enabled to allow direct x overlay windows etc. Worked first try through lutris though and ive been playing the game on and off for the past year through that without issues.
The Elder Scrolls Online, works perfectly fine through proton. You can even get the standard addon manager (minion) working through standard wine and just point it at the right folder for addons to work. A few that require running programs to update stuff will be fiddly though. (Like tamriel trade centre)
Final fantasy XIV has an unofficial launcher (XIVlauncher) you can snag up on the flathub that makes installing and running the game easy. I have not played the steam version, but i imagine it works as well.
Warframe works fine through steam, but its barely an MMO.
Second life works fine with the common viewer, firestorm, having a native linux version.