For example, I'd like to fork a GNU Savannah git repo, and use GitHub's pull request button to signal GNU Savannah that I'd like to submit a patch. Is this possible?
No, pull requests are something in the project management layer, in this case the GitHub website. It has nothing to do with Git itself.
A pull request in GitHub is just a ticket with some patches attached to it. You could create yours by having your changes formatted as patches using
Even if you mirror the repository to GitHub, you don't have a target repository to send the pull request to. For pull requests on GitHub to work, your and the target repository have to be on GitHub. Note that this does not exclude the (unlikely) possibility to provide both systems (GNU Savannah and GitHub) to be actively maintained by the project maintainer.