This is a question in the FAQ in the README.txt for 7-dev:

Q: Why the connect button does not appear?
Q: Facebook Connect dialog says: "Invalid Argument: Given URL is not allowed by the Application configuration".
A: Check 5th step of installation instructions. Check for error messages on page admin/reports/status.

Step 5 is:

5. Configure the module through admin interface (admin/settings/fbconnect) using the information provided by Facebook
(Application ID, Application Secret).

(BTW the URL for the admin interface is wrong: admin/settings/fbconnect is for Drupal 6. In Drupal 7 it has moved to admin/config/people/fbconnect.)

Well, I did have the problem that the button didn't appear, but the solution could not be found in step 5.

After wasting a few hours I found that the Facebook login block simply was not configured to appear! And managing behavior of this block is not covered in this admin section (which would not be a bad idea by itself). I don't even know if it's normal that I had this problem. Maybe it was caused with the way I installed the module; I needed the updates from the last few days so I got it via git.

So, I suggest to at least add this paragraph to the FAQ answer:

Make sure that your Facebook login block is placed on the page: see admin/structure/block, or the advanced configuration interface at admin/structure/block/manage/fbconnect/fbconnect_login/configure.

Perhaps it would be possible to have the installation automatically add this block in the same region as the User login block for the various themes? (admin/structure/block/manage/user/login/configure) Because that would achieve what most people probably expect.