As long as the original author has clicked the checkbox in the bottom right:
If that box is checked, then you can push back to the original branch without needing to add a remote by using:
git push git@github.com:user/repo local_branch_name:remote_branch_name
This is particularly useful if you’re using a tool like hub where you can check out a pull request without needing to add a remote.
Edit: You also need to have write access to the repository where the PR has been opened.
Further Edit: The new command line tool from GitHub does a good job of checking out PRs so that you can just use git push as is to push updates to remotes (assuming you have permission).