I've recently starting seeing the above error with ever-increasing frequency on our build server. Nothing has changed in our TeamCity configuration during this period, so I'm guessing it might be changes at GitHub that are causing the error.
I've tried changing our VCS polling interval from 60s down to 600s in case GitHub was doing some kind of connection throttling, but there has been no affect.
Is it possible to make TeamCity less sensitive to connection timeouts?
For proof, try doing this in your server, where TC is installed.
(command line, of course)
C:\git\bin\git.exe clone https://github.com/libgit2/libgit2.git
and this should not work most of the time.
So AZURE has a networking bug and they know about it and are trying to resolve the issue.
This info was provided via GitHub after they worked with Azure to figure out what was going on.
You have to use SSH KEYS as a current workaround.