Is this permanent or still some workaround?
Right now when I use the google search with the parameter
site:forums.mp3tag.de
then it works which means: I get hits that claim to link to pages on https://forums.mp3tag.de
but actually get transferred to https://community.mp3tag.de
While when I use
Site:community.mp3tag.de
Google does not find anything.
Will this stay like it or is it just a question of time until the google crawlers have updated their cache and catalogued the new address?
Well, as @Florian uses 301 redirect, Bots just need to recrawl the whole site.
Eventially, when everything has been recrawled, Google will phase out old subdomain from index and this "issue" goes away.
Redirecting via server side 301 is Google advised method anyway: https://support.google.com/webmasters/answer/93633?hl=en
From my personal experience, it will take maximum one month if you have huge forum, so this one should take considerably less.
Also you (if you are site owner) can use Google webmaster tools, to help Bots out and make transition faster. 
Exactly as @Etz described. I'm using HTTP 301 redirects so the old subdomain should slowly disappear from search engines.
In addition, I've added redirects for every old forum URL to the new forum URL, trying to make this transition phase as smooth as possible.