That's a new one to us. We'll agree, it seems as though you can't connect to Facebook itself, which is obviously strange. We're not seeing that on any of our development systems, so it doesn't appear to be an outage with Facebook Connect, which does happen (though rarely).
If you have SSH access to your server, can you try typing the following commands, and let us know what you get? You should see something similar to what's posted below. This verifies that your server can 'see' Facebook. If you don't have access, you may need to contact your hosting provider to see if there's any reason your site can't access these addresses.
Command:
nslookup api.facebook.com
This verifies that your site can lookup the actual IP address of the Facebook API site. Should see something like:
Non-authoritative answer:
Name: api.facebook.com
Address: 69.63.181.48
Command:
This verifies that you can send and receive from that address. If this fails, would indicate a firewall or other security mechanism preventing communication. Should see (Press cntrl-C to stop the 'ping'):
PING api.facebook.com (69.63.181.46) 56(84) bytes of data.
64 bytes from api-11-01-snc2.facebook.com (69.63.181.46): icmp_seq=1 ttl=245 time=46.7 ms
64 bytes from api-11-01-snc2.facebook.com (69.63.181.46): icmp_seq=2 ttl=245 time=46.8 ms
64 bytes from api-11-01-snc2.facebook.com (69.63.181.46): icmp_seq=3 ttl=245 time=46.6 ms
There are other possibilities regarding server/site configuration issues, but that error seems to indicate everything's good on your site's end... but when the request goes out to Facebook, it's just going to a black hole...
Let us know if this helps, or is way over your head. Unfortunately, the next step would likely be to contact your hosting provider.
Thanks!