You will need to install a user script manager extension to install this script.
Since the OpenStreetMap website was lacking a notification feature for the user diaries, this was a way to improve this situation to some degree. But even now this script still has the advantage to work without having to log into the website and might be sufficient for checking the latest diary posts.
The detection of new comments is implemented by storing and comparing the language dependent comments link text. So it doesn't work if switching the website language. In this case, all comments will appear to be new again.
Source repository: https://github.com/marcows/osm-diaries-toc