There's 2 different issues I now see on both of those sites. I don't know if they're the cause, but it's where I would investigate first.
On the Joomla 3.1.4 site, there is the following Javascript error on that page:
Uncaught TypeError: Object [object Object] has no method 'tooltip' at line 149
The code at that line is:
<script type="text/javascript">
(function() {
var strings = {"JLIB_FORM_FIELD_INVALID":"\u00c9rv\u00e9nytelen mez\u0151: "};
if (typeof Joomla == 'undefined') {
Joomla = {};
Joomla.JText = strings;
}
else {
Joomla.JText.load(strings);
}
})();
</script>I'm not sure what that is doing, but the Javascript is throwing an error which can prevent further Javascript from functioning.
On the 2.5.13 site, you look to have Gavick's Facebook integration also enabled. You'll need to go into the template settings and disable their Facebook integration to prevent the double inclusion of the Facebook Javascript library.
The above are both the next suggestion as they are both definitely problems. Fixing those may unearth something else, but I don't see any other issues with either site after taking a pretty thorough look at the HTML.
If you'd like to Private message me admin credentials to either site, I'll gladly look at the configuration further. On the 2.5.13 site, I could help disable the Gavick FB integration. On the 3.1.4 site, I don't know where that bad Javascript is coming from, so I can't really help you fix that.
In general, JFBConnect is used on many, many sites with Gavick templates. They can cause issues, but we know that the 2 can play together just fine.
I hope that helps,
Alex
Cant get facebook login to work
