And the move is done, thanks to Erlend from the Discourse team!
I had to update the settings at GitHub, Twitter and Google to fix the social logins and tell Google that the new location trying to log in via POP for getting reply e-mails was legit, but now everything seems to be working.
I've updated the forum URLs in the navigation bar and on the Tiled website and set up a redirect from
forum.mapeditor.org which will activate as soon as the DNS settings have propagated.
A big thank-you to the Discourse team for providing this service!