I've put together a Heroku buildpack to make it easier to run the OSRM routing API (
osrm-routed) on Heroku - it's available on Github.
There are a few caveats - mainly about the size of data files that can be used - more detail is in the README.
A note on memory management: It doesn't look like it's possible to use
osrm-datastore and shared memory on Heroku - I'm not sure if this has any implications on running the OSRM daemon on Heroku in a Production environment.