I am unable to get this working. On a different web hosting service, I was able to set the Content-Encoding header to gzip, but I've looked everywhere and don't see a way to set headers on Heroku. The app is failing because the file isn't being decoded (it's a gzipped JSON file). I am looking at the network tab on Chrome and the response doesn't include the Content-Encoding header.
Heroku is an application host (a Platform as a Service), not a web host. The Heroku platform as such is not an HTTP server. Your application defines its own server infrastructure based on your source code, dependencies, Procfile, and the combination of buildpacks you select to run in Heroku.
The behavior of how your static files are served is defined by your application and you'll need to triage and resolve it based on the specific stack you are using. You won't be able to set headers in the Heroku platform itself.