Putting your database somewhere on the cloud won't eliminate the need for you to manage it. Heroku gives you a database when you host an application with them, and they'll set it up and make sure it works for you.
I would recommend keeping the database on the same system that the application is running on (in this case, Heroku). This gives you a few benefits over cloud hosting, such as:
- Low latency means faster requests
- Support; Since Heroku is hosting the database, you have access to their support to fix any issues.
- Easily upgradeable; If you need to scale your database, Heroku can manage it for you (albeit at a cost increase)
Cloud hosting services are only needed for high stress, high traffic systems where you may need to spin up new slaves to handle requests for you.