Lock the task status when reaches the Complete status

You could create a Project repository for these tasks / items. Simply move them to the repository once  they're complete.  Just set the repository Status = Complete after the move to ensure no one can edit the tasks/items.  

With standard WF functionality, I don't believe there's any automatic way to accomplish this. If I had to do it by hand, I could update the sharing settings on the Task/Issue to only allow View access to the original assignees, revoking the Contribute/Manage access they had previously. 

If Fusion is an option, then you could do that as I described above using the closure of the task/issue as the trigger for the scenario. I have one like that in my instance that detects the closure of an Issue to auto-convert it to a Project, and then I reset sharing security on the originating issue so no-one can make edits there instead of the project.