In order to run an arbitrary script on Heroku, you can use the Heroku console. They state that "you can use heroku console as a stand-in for Rails’s script runner", but one key component of the script runner is that you can run files. You can run files in your git repository that are deployed to the instance that you want to run them on by using the method advocated by Steve Wilhelm.
What if the file is not checked in or you need to make rapid modifications for prototyping? If you have a lot of code that you want to write and you want to be able to easily modify it, try the solution below. Manually typing in many multi-line statements at the console (and retyping when you make an inevitable mistake) is frustrating.