This is currently not possible. There is a feature request and someone already started working on it, but that work has stalled.
Currently the only option I see is to open a new merge request and link the old one. Probably not what you want, but easy to do, and at least the two MRs are linked.
I also looked at the API, whether it is possible to duplicate the old MR. While the API does allow you to add discussions, this is very limited. Especially as it is currently not possible to add discussions to a specific file and line number. It is not even possible to extract that information via the API.
Thus the only option to accomplish "reopening" or "duplicating" a merge request is to mess with the database or work on the merge request.