You could, but you'd have to have some intermediate service to handle that process. One great choice is Travis CI[1] which is really a continuous integration service, but would let you do what you want. You can also just attach multiple URLs to a single git remote so you push to both GitHub and Heroku all the time.
However, I think running a test suite before deploying is a much safer option as it stops you from accidentally deploying something you haven't properly tested.