How would I configure TeamCity to support a GitHub pull-request model for source-code integrations?
You can monitor pull-requests by teamcity: http://blog.jetbrains.com/teamcity/2013/02/automatically-building-pull-requests-from-github-with-teamcity/
To my knowledge TeamCity will not "seek out" your github forks in any automated fashion.
I would solve this by creating a vcs root for each developer's forked repo, and the mainline repo. Then attach each of those roots to the build. Docs.
Unfortunately it is a manual process. Once per developer, per component repo. If I really wanted to automate this I'd look at a combination of Github Hooks and adding vcs roots via the TeamCity REST Api.