I having the same issue as OP (Facetime opens, has me sign in and shows me my email addresses, I click next, and then goes back to the login screen).
All has been running fine for months. Never used Facetime before but a family member wanted to call me and I only just discovered the issue.
I am running Acer 4830T booting 10.9.2 by Chameleon 2.2svn (svn-r2377). Spec is Intel Core i5-2410M CPU @ 2.30GHz with 4GB memory.
I changed my password for AppleID but made no change.
SMBIOS.plist was generated by Chameleon Wizard with random serial number.
There's an NVRAM.uuid.plist file in /Extra and I used the uuid part of that filename and created a SMsystemuuid key in SMBIOS.plist but that made no difference.
Any other suggestions ?