Not sure about the full requirement. Do you want to delete them manually since its stucked in User's inbox? or the requirement itself is to delete the task after user's completes the task?
with user folder, you mean: 'Task to Track' or 'Task to Perform' folder?
I would recommend review the workflow process design (possible issue with signoffs) if you observe that the tasks from a particular workflow continues to linger in user's folder after the workflow is finished.
I've seen this behavior till Teamcenter 10.x.
Are you on TC10.x or less?
When this happens, typically it suggests that either the rule handler or action handler has encountered an error condition. That is - either rule handler returned EPM_nogo, or action handler returned other than ITK_ok.
Your post suggests that the workflow seems to be finished, but does not leave the inbox. This would suggest that error is in complete action of the root task of the workflow process. Look in all action handler/rule handlers there. syslog/jnl file may be able to give you more information if error is reported there.
If you don't have any action/rule handlers in complete action of the root task, then check handlers in complete action of the last task prior to finishing the job.
If handlers are encountering error conditions, then you will be able to see those errors, if you select the task in the inbox, right click, and go to action=>complete (perform manual complete action on the task). It should display error message if handlers are reporting it properly. That should give you some idea about what's happening.