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
https://devcenter.heroku.com/articles/platform-api-reference#build-create

There is a Builds CLI plugin that makes this easier:

heroku builds:create --source-url https://api.github.com/repos/<username>/<repo>/tarball/<branch>?access_token=<github-personal-access-token> --app <app-name>

How to generate a GitHub personal access token.