How do I process inbound e-mails with Mandrill on Heroku?

I think you need to look at Mandrill documentation on processing inbound emails.

From this, it looks like there is nothing special about doing this in Heroku - as long as you have a web page in your site to handle the inbound message, all the configuration is done with the Mandrill dashboard.

Have you checked out the documentation on using the Mandrill add-on with Heroku? It's available here with examples for multiple languages:

https://devcenter.heroku.com/articles/mandrill

Using Mandrill directly with Heroku is quite easy. You only need to set a STMP username and password in Mandrill within: Your_Mandrill_Account >> Settings >> SMTP & API Credentials

Host smtp.mandrillapp.com Port 587 SMTP Username your_userSMTP Password your_pass

You also need to create a new API. Add a domain, and create or reuse a template for your emails.

Once you have done that, the only thing you have to do is use your create your email within your application and send it using the mandrill settings you've just created. Take a look at the Mandrill API https://mandrillapp.com/api/docs/messages.html#method=send

There is also some projects in Java that works as a wrapper so that is even easier to user Mandrill. https://github.com/cribbstechnologies/Java-Mandrill-Wrapper/blob/master/src/it/java/com/cribbstechnologies/clients/mandrill/it/MessagesTest.java. I suppose that there will be examples in any language you may be using