I know a few people seem to have solved this (pg 9 of this thread), but I wanted to add my two cents. I first want to say that apple was useless as everyone else noticed and I am positive it has nothing to do with an app or software that I downloaded. I'm in another country and wasn't even able to download anything until post Facetime shutdown. (that was a whole other issue I had to solve)
After several hours of troubleshooting and relearning how useless Apple support is, I believe the only way to get Facetime working again is to restore your phone to factory settings. There is software out there like Wondershare that will allow to decode the backup file and pick and choose what you put back on your phone without having to apply the restoration, but obviously those cost money.
Some items I accomplished beforehand:
-Download the app "My Contacts Backup" and email yourself your contacts because you will lose those. You can then email yourself a .vcf file that you can then use to restore your contacts (If you have it synced to outlook, you can use that file too)
-Save you photos to another source: You will lose these too
-Email yourself any notes you might need
-Sync your phone
-Back up your phone to both the cloud and your computer (just to be safe)
To restore: Settings>General>Reset>Erase All Content and Settings
Restore it as a new iPhone
Once reformatted, you can open the setup your email, open the .vcf file, transfer your photos and sync your music/apps/movies etc
I did this on both mine and my wife's phone. One thing I did not try because I really didn't need to, was restoring the backup you previously made after restoring as a new phone. My guess is it won't work, but feel free to try it.
Our Facetime now works. (We have not downloaded iOS7 yet.)