If you want to log out of the Game Center in iOS to play another Clash of Clans account, here is the way to do that!
Just following these simple steps. I have just tested it on my iPad with iOS 7 and iPhone running 8 but I think it works on all iOS versions!
On your iOS device, open Settings menu then choose the Game Center menu.
Tap on the Apple ID (yellow line in the above image) menu then choose Sign Out.
You have done.
After logging out, you can use another Apple ID to Sign In. Sometimes the new Apple ID will load the old data from the previous account. It is recommend that you should restart your phone after logging out of Game Center.