Can I remove quests from the list on the right side of the screen?

[Niwashi]

I wish we could. Apparently it was possible at one point during beta but had some bugs and the feature got removed rather than fixed. (I thought I'd seen a dev post somewhere saying they were hoping to fix it and add the feature back in, but I can't find it now, so it's possible I'm mis-remembering from one of the player posts I've seen requesting it.)

[KagatoAC]

It was indeed in the Beta right up until release, there was a checkbox that allowed you to decide which quests were displayed. A lot of people were surprised when it went to release and it couldn't be found anymore