I recently needed to run Grails database migrations on Heroku and used the following steps to run these.

First,  the Grails wrapper is required which can be installed using the wrapper command.

After running the wrapper command for the first time on Heroku, it complained that JAVA_HOME needs to be set. So, I used the following command to determine JAVA_HOME:

$ heroku run which java
Running `which java` attached to terminal... up, run.6852
/app/.jdk/bin/java

I used just the home part of the path to set the JAVA_HOME variable:

$ heroku config:set JAVA_HOME='/app/.jdk'
Setting config vars and restarting ... done, v22
JAVA_HOME: /app/.jdk

Next, I ran the migration command with the environment I required:

$ heroku run ./grailsw -Dgrails.env=production dbm-update
| Finished dbm-update

Finally, I confirmed the tables were created that I needed:

$ heroku pg:info
=== HEROKU_POSTGRESQL_PURPLE_URL (DATABASE_URL)
Plan: Dev
Status: Available
Connections: 0
PG Version: 9.3.3
Created: 2014-05-11 03:38 UTC
Data Size: 6.6 MB
Tables: 3
Rows: 3/10000 (In compliance)

Using the above  steps, the Grails wrapper command could also be used to run other Grails commands on Heroku.