The iOS 7 update has caused many users to not be able to sign into FaceTime or iMessage on their device. Although each incident appears different I have compiled a list of possible solutions that users have reported successful. Please be aware the most users had success with these solutions but some did not.
Fix Login Issue with FaceTime & iMessage Solution #1
1. Settings > General > FaceTime > Turn Off
2. Settings > General > iMessage > Turn Off
3. Settings > General > Reset > Reset Network Settings
4. Connect to your WiFi
5. Turn on FaceTime & iMessage and connect.
Fix Login Issue with FaceTime & iMessage Solution #2
1. Settings > General > Reset > Reset All Settings
2. After the reset, connect to FaceTime and iMessage.
Fix Login Issue with FaceTime & iMessage Solution #3
1. Settings > Wi-Fi > Click on your WiFi Network
2. Change the DNS to 18.104.22.168 and try connecting to FaceTime & iMessage
Fix Login Issue with FaceTime & iMessage Solution #4
1. Settings > General > Date & Time > Turn on ‘Set Automatically’
Connect to FaceTime & iMessage
Fix Login Issue with FaceTime & iMessage Last Solution
The last solution is to do a full restore. I would not recommend this method as this will delete everything on your device. At this point, if all the other solutions did not work I would recommend that you wait until Apple releases an update to resolve the issue. This is a well known issue so I am sure an update will come soon.
Please comment and let us know if you had any success with any of the solutions. I was able to get it working using the first solution.