In general, no. Once a user completes a course, it will stay "completed" unless manually modified. However, if a course is configured with a Renewal Interval, the user's completion status will reset on the date selected on the Content>Courses>Course Details page.
Note: For online courses, if you simply update the course with a new SCORM package, a user's completion information will not be changed.
This article is also included in Learning Path Development>Creating & Uploading Courses and User Management>Tracking User Progress