Can I "archive" or hide completed submissions?

Once a submitted form is processed, we would like to mark it as complete and move it to a finished folder or something.  Is there a way to do that?

As far as I know, there is no way to archive them in another folder, the forms are sorted by order of arrival, you can even click on the gear icon, and sort by time frame:

Wha you can do is to integrate the form with Dropbox, or with Google Drive, and move the submissions manually to another folder within your Dropbox or Google Drive account.