Thank you everybody for the answers. @Anthony, the coloured right hand area is visible only if you go to the task, right? "For us, the Task Owner is not an important feature at all so we'd like it just to disappear from everyone's My Work list when the task is complete." Same for us...

In "My Work" in Workfront, in the "Working On" tab, I have tasks that I committed to, but which were closed or cancelled. I just didn't click on the "Done" button. Is it really necessary to click this button or can I automate this somehow?.