When did you create your Facebook Application? If recently it could be due to the following.
forge.joomlapolis.com/issues/4081
Basically on July 10th Facebook will be making breaking changes to their API and this could be breaking the invite feature if you created your Application recently as it'd already have force enabled the breaking changes. Will need to migrate my Application and test again to see if this is the case. Edit your Application at Facebook and click the "Advanced" menu item on the left then under "Migrations" let me know what is set as "Enabled".
It seams others have also reported this issue to Facebook in the below Facebook bug ticket, but it says they fixed it. Seams to be an issue on Facebooks end.
developers.facebook.com/bugs/284149695026014/
Please note Facebook will not work on localhost properly. Your site needs to be accessible to the internet. So ensure you're testing on a live domain.