Heroku has a readonly file system. The directory where you can write are ./tmp or ./log. You can't write inside the public folder. That's because of how they manage their dynos and the way to scale them. If you want to store something, use the ./tmp or, recommended, a s3 bucket. (as I presume 'tmp' stands for 'temporary' :D)
More info here: https://devcenter.heroku.com/articles/read-only-filesystem