Hi Rachel,
I'm not sure what version of Moodle you're running, but if you're on 2.3, then check and see if the "Prevent Late Submissions" option in the Assignment settings is set to "No." If it is set to "No", then Moodle will still think that students can submit assignments past the due date and will leave it listed in the Course Overview block on the My home page. Changing this option to "Yes" should remove the assignment from the list after the due date. (Some instructors like to leave the "No" so that students can turn in things late, however they don't always realize how this impacts the My home page.) It sounds like nothing is submitted online anyway, so this setting should be an okay setting for your purposes.
Also assignments that only have an open date without a due date will remain in the Course Overview list, regardless of the Prevent Late Assignments setting.
As an FYI, you can test all of this pretty quickly yourself by setting up your own course, enrolling yourself and then adding a few assignments that enable different combinations of these settings (start date, due date, prevent late submissions). The combinations all have slightly different functionality in the Course Overview, Calendar and Upcoming Events blocks, so it's worth looking at yourself to see their impact.
HTH!
Justina Hohmann