How to reactivate controller Binding of Isaac Rebirth
Solution 1:
If you have the latest update, you can go to your Isaac folder, look for options.ini and open it with a text-editor of your choice. set PauseOnFocusLost to 1 (PauseOnFocusLost = 1).
Now you can simply use Alt+Tab or any other shortcut which changes the window focus and the game will lose focus, triggering a pause.
This way you don't need to unplug the controller. Also most coontrollers have a pause-key in Rebirth. You can resume the game using the controller with the key for options-back (the same you use to go back one level in menus).