The "You have assignments that need attention" notification seems to be perpetually present for both students and teachers, no matter what the student/teacher submits, accomplishes or grades.

For example:
-> even if a student has completed and submitted all assignments for a given course, he still gets the "You have assignments that need attention" message for the course on his home page
-> even if a teacher has graded all pending assignments for a course, he is still getting the "You have assignments that need attention" message for the course on his home page.

Related discussion here:
https://moodle.org/mod/forum/discuss.php?d=226744