The app I am currently hosting on Heroku allows users to submit photos. Initially, I was thinking about storing those photos on the filesystem, as storing them in the database is apparently bad practice.
However, it seems there is no permanent filesystem on Heroku, only an ephemeral one. Is this true and, if so, what are my options with regards to storing photos and other files?
It is true. Heroku allows you to create cloud apps, but those cloud apps are not "permanent" - they are instances (or "slugs") that can be replicated multiple times on Amazon's EC2 (that's why scaling is so easy with Heroku). If you were to push a new version of your app, then the slug will be recompiled, and any files you had saved to the filesystem in the previous instance would be lost.
Your best bet (whether on Heroku or otherwise) is to save user submitted photos to a CDN. Since you are on Heroku, and Heroku uses AWS, I'd recommend Amazon S3, with optionally enabling CloudFront.
This is beneficial not only because it gets around Heroku's ephemeral "limitation", but also because a CDN is much faster, and will provide a better service for your webapp and experience for your users.