How do I force a new deploy without adding a commit to my Github repo?

Issue

You want to re-deploy the same commit SHA from a private GitHub repo.

Resolution

This can be done with the builds-create Platform API endpoint.

There is a Builds CLI plugin that makes this easier:

heroku builds:create --source-url https://user:token@api.github.com/repos/<username>/<repo name>/tarball/master/ --app <app-name>

Where user is your Github username and token is a Github personal access token (reference)