I want to be able to upload images to a server through something like a RESTful API, which then processes the image using an ImageJ script and then returns numerical results (no image is returned)
Can this be done easily? and can it be built on Heroku?
ImageJ works well headless; see http://imagej.net/Headless. So you could certainly build a server application which uses ImageJ based on any of several different Java server platforms (JBoss, Glassfish, Jersey, etc.). Especially if you leverage the new ImageJ2 API, there is a good separation of concerns which would allow you to build an effective client-server application.
To my knowledge, however, no one has created such a thing yet. I started playing with the idea using Jersey, but there is nothing really there right now, nor any time to work on it. Contributions in this area would be most welcome!
See also the ImageJ-OMERO project, which exposes ImageJ commands (including scripts/macros) as OMERO scripts which run on the server side.