When I launch Sourcetree, its UI immediately shows:
Authentication Completed with an Error
An error occurred while authenticating your account ...  You can view the server's response below.
Error Contents
message: Requires authentication
https://developer.github.com/v3/users/#get-the-authenticated-user
Sign In
 
When I click the Sign In button, a new dialog appears:
Please sign in to this account:
Sign in to Github to continue to Sourcetree for Mac
Username or email address
Password
Sign In

If I enter my username or password incorrectly and click Sign In, it gets rejected with "Incorrect username or password" (testing that logging in with the correct username and password will actually work).

When I enter my username and password correctly and click Sign In, it gets accepted, the dialog disappears, and then the original screen starting with "Authentication Completed with an Error" reappears (see top of this report).

This much may have already been reported, but I couldn't find any mention in the open bug list of the next feature - the program can't be closed. The red "X" in the upper left corner of the Sourcetree UI is gone, and if I click "Sourcetree", "Quit Sourcetree" in the top menu, it beeps at me and the program continues running. Same result if I attempt to quit it from the dock menu. The only way to close it is Force Quit with Cmd+Opt+Esc or force a restart of the Mac.

Finally, if I open a terminal and do a pull from Github, it works fine.