There's a couple of strange things I'm seeing. First, in Chrome, if I go to:
http://www.xn--ditazubi-1za.de/
I stay on that page.
If I go there in Firefox, I'm redirected to:
www.diätazubi.de/
On that page, in both Chrome and Firefox, when I try to login, Facebook's Javascript API is throwing the following error:
Error: Domain contained invalid characters.
Source File: http://connect.facebook.net/de_DE/all.js
Line: 46
I don't know why, but for some reason, it doesn't like your domain. I see you have a UTF-8 character in there, but that's not terribly uncommon. I don't know if this is a change with Facebook itself, or a bug that's specific to your site. We'll have to investigate that side more, but can you explain why the redirect is happening only in some browsers and what the correct URL that you'll be using is?
Also, do you have different API keys set for both sites? Each API key will only work on one domain. It's possible you're using the wrong API key on the site that isn't working.
Thanks,
Alex