The short answer is no. You cannot delete or rename an open file and the file would have to be open in order to run the macro. The best solution I can think of is to put the new templates somewhere the users can access them and add the following code to the ThisDocument module of each old template (change the due dates as appropriate) and save as macro enabled templates. Users will not then be able to open the template or create new documents from it from that date, but would see a message instead.
Code:
Option Explicit
Private Sub Document_New()
If Format(Date, "yyyymmdd") >= Format("07/07/2017", "yyyymmdd") Then
MsgBox "The template has expired, please download the latest version from 'somewhere'"
ActiveDocument.Close
End If
lbl_Exit:
Exit Sub
End Sub
Private Sub Document_Open()
If Format(Date, "yyyymmdd") >= Format("07/07/2017", "yyyymmdd") Then
MsgBox "The template has expired, please download the latest version from 'somewhere'"
ThisDocument.Close
End If
lbl_Exit:
Exit Sub
End Sub
See attached example