In 10.1 beta (14B72), my game center connection logs me out and fails to reconnect until I reboot of the device (iPhone 6S+). When the lockout occurs, I go into settings and select the "Game Center" setting, it locks up the settings app entirely - the "Game Center" selection remains grey to indicate my selection, but the view never changes and I cannot select or scroll the settings view. I can return to the home screen, but then going back to settings brings up a blank view (apparently its still locked up in GameCenter).
Eventually it appears to timeout or crash the settings, after which the settings can be reopened successfully, but game center remains unable to connect.