Client JS + Django Rest Framework
Heroku is a application server platform. It's not really designed to serve static code.
The approach I've previously taken is to build the Django part normally, and have a single view that serves out a bootstrap template for everything under the JS App root.
Say I have a {Angular,Ember} app living at mydomain.com/app/, then everything under that will serve the bootstrap template (which includes serialised values queried from database) and calls the JS boostrap method to startup your app, and then the app takes over routing from that point, and renders out it's views.
At this point, all the data for the views is coming from django-rest-framework/django-tasypie.
Using this method and leveraging django-pipeline & django-boto's S3 storage backend, You should be able to serve a decently sized project with Django & {Angular,Ember}