How do I disable "An update for GNOME Shell extensions" notifications?

To get rid of the update notification, you will need to disabling the automatic check for updates to Gnome Shell Extensions in preferences of the "Gnome Shell Preferences" extension of your browser. In Firefox: - Select Tools - Add-ons (or "Add-ons" under the Open Menu button) - Select "extensions" in the menu on the left, then click the "Preferences" button next to "GNOME Shell Integration". - Scroll to the bottom of the page, where you can disable the option "Check for extensions update".

The automatic check does do only that: check whether there is an update and notify you. If yo do not want the notification, then it does not make sense to check. So deleting the notification means you need to disable the check.

There is currently no mechanism where updates to Gnome Shell extensions that have been installed by the user are automatically installed. In contrast, extensions that have been installed system-wide will be automatically updated like any other software through your package manager.