[Plugin: Simple Facebook Connect] Publish function doesn't work when the plugin is installed on

Did your URL change? If so, then you’ll need to go edit your FB Application and correct it to have the new URLs.

Nope, the URL didn’t change, that’s why I’m a bit confused.
I didn’t re-auth the plugin as it already says that the permission has been granted.

The “message about a problem with the plugin” on the edit post page is almost always having bad URLs in the FB Application. But other plugins can cause it too.

Could it be because the network has another URL and the site’s URL is mapped?

Nope. The URL that displays in the browser is the one that matters.

Now, if the admin area has a different URL than the main site, then that will indeed break the thing on the Edit Post pages. The URL must match for it to work. This is due to how Facebook does cookies.

The admin area is slightly different due to the use of SSL. Could it be it?
I’ve tried to fix the problem by using HTTPS for WordPress, but it didn’t seem to work last time I tried.

SSL might be the issue. It tries to handle that case, but I’m not sure it does it correctly.