Sorry, I originally thought this was for a form. Yes, you can clear out the drop-down selection using the Assign Field Values activity in workflow. For your situation, after the form is moved to the specific user's folder, you can use Starting Rules in workflow to trigger a workflow to search the repository for that document and use the For Each Entry (assuming there could be multiple documents) and the Assign Field Values. In the Assign Field Values, select the field that contains the user(s), but don't add a Field Value.
Depending on your setup of how the documents reach the personalized folder and how they may be accessed will determine your starting rules and continual flow, but that activity will remove the name(s) that are assigned when the document hits that folder initially.