Hello,
i have a problem after changing my webhoster. The site works great but not so forum. 
I get logged in but after clicking to a thread i get logged out… The same problem occurs after writing a post.
What is it??
Here is a login:
login: test
pw: test
http://www.pommes-rot-weiss.com/forum/
Maybe you have an idea. Would be great 
Could it be this:
https://simple-press.com/documentation/codex/faq/troubleshooting/forum-displays-wrong-information/
Then something you have installed there must be playing tricks. Whether you are logged in or logged out is nothing to do with Simple:Press – this is a WordPress process. To discover whether the current user is logged in or not – we just accept whatever WordPress tells us.
So apart from your web host – what else has changed? And if the web host is the ONLY change then it might also be worth asking them. And remember – your login status is for the site NOT the forum.
Additional
If I log in to your site in your home page then I remain logged in through forum page loads. It seems then that you have something that is overriding the forum login form – which simply calls the WP login process.