List remote refs failed: java.net.ConnectException: Connection time out: github.com

I've figured out the answer.

TeamCity has no issues - it's actually AZURE that has a problem.

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.

Conclusion

You have to use SSH KEYS as a current workaround.

We have made it less sensitive: now git-plugin will retry an operation in the case of ConnectException. To get this behaviour you need to install the latest build of git-plugin from teamcity.jetbrains.com. To do that put the zip from the build's artifacts into <TEAMCITY_DATA_DIR>/plugins and restart the server.