Q: Face time stopped working iPhone 6

Did you already try to reset the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again?

You will not lose data by resetting.

If this does not work, switch FaceTime off and on again.

If still no luck, set it up as new device:

How to erase your iOS device and then set it up as a new device or restore it from backups

Also check this again:

If you can't make or receive FaceTime calls - Apple Support

disconnect Facetime in Settings and reconnect.

happens from time to time.