Hi dryabov,
Nope unfortunately I didnt fix the issue at all (I tried and tinkered but it alluded to nothing)...
The problem is that it does indeed do something when you click on it (it reloads the page), but it fails to log you in (normally there is a pop-up that says that you are being logged in automatically).
I have disabled ajax navigation and fast click and it is still the same. I have tested it on Android and somebody else (not sure they were really testing it... I think they were simply button baching...) tested it on Iphone and neither works.
I am currently using the Elegance template and the Kunena template (both of which look awesome btw).
I am sure that it has to do with the Elegance template. If you scroll down, and click on "standard version", you can log in using the standard version "FB Login" and then are logged in (even if one switches back to mobile version).