Hosted by 3rd parties
There are several third parties where you can run a Firefly III instance. Keep in mind that there's no such thing as a "free lunch", and these options are either paid or severely limited (very slow or can't handle many transactions).
Considerations when using Heroku
Heroku uses what is called an "ephemeral file system" and it will not be able to store attachments. They will be deleted every day. Don't use Firefly III on Heroku in combination with sensitive or rare file attachments.
The free-tier database can hold a maximum of 10,000 rows, which is about one year's worth of transactions.
Firefly III is featured in AMPPS. You can download AMPPS for Windows, Linux and Mac and Firefly III will be available as a package there.
Firefly III is featured on Yunohost.