We started with a free tier (copying the Heroku model) and had to fight with 'free users' as well.
The free tier created a lot of noise distracting us from our paying customers. Support requests from users with no interest ever upgrading, often beginners and noobs. Cat and mouse games with script kiddies trying to abuse the service. Fraud and phishing … Most free tier Apps are just tests, maybe there is an index file printing a hello world. Yet you need to keep some responsibility to keep that data around.
After a while we changed it to free trial (try before buy). That works much better for us.