Hey,
I had the same issue.
The issue is kind of the same people had with facetime at the release of OSX 10.7
The issue was related to custom entries in the file /etc/hosts
Check if you have edited the file /etc/hosts (using terminal)
If you did, temporarily remove all custom edits from it till it looks like this:
_____________________________________
##
# Host Database
#
# localhost is used to configure the loopback interface
# when the system is booting. Do not change this entry.
##
127.0.0.1 localhost
255.255.255.255 broadcasthost
::1 localhost
fe80::1%lo0 localhost
______________________________________
Save the file and afterwards run the following command in terminal: dscacheutil -flushcache
After this try logging in to Game Center again and it should work.
Once the login in succesful, you can reapply the custom settings in the /etc/hosts file. Game Center will continue to work.
Hope this helps.
With kind regards,