I'm not sure of the exact cause, but it seems to be something related to your template. Facebook relies heavily on Javascript, and if there are any errors on the page, it can prevent it from running normally. When I load your page, I see the following Javascript error, which appears to be generated from your template:
Uncaught TypeError: Object #<HTMLUListElement> has no method 'init'
I believe, because of this error, the Facebook Javascript library isn't being initialized properly, which then won't let the user login. As a way to test this, by simply adding ?template=rhuk_milkyway to the end of your URL, you'll load the default Joomla template and see that the Login button does work. This means it's either the template conflicting, or something that only loads in a module position that exists in the template you're using.
Can you try creating a very stripped down page with no modules loaded and see if the Login button works then? Alternatively, if there are any Javascript/flashy features that you can temporarily disable to see if that fixes the problem, it will definitely help us narrow down the issue further.
Hope this helps get you started,
Alex