To drop the database, if you are using SHARED_DATABASE_URL:
heroku pg:reset DATABASE
To recreate the database with nothing in it:
heroku run rake db:migrate
To populate the database with your seed data:
heroku run rake db:seed
You can combine the last two into one action by executing this:
heroku run rake db:setup
Edit 2014-04-18: rake db:setup doesn’t work with Rails 4, it fails with a “Couldn’t create database error”.
You can do this with pretty much any rake command, but there are exceptions. For example, db:reset doesn’t work via heroku run rake. You have to use pg:reset instead.
More information can be found in Heroku’s documentation.