When I connect pgAdmin III to PostgreSQL on Herkou it says: "The server lacks instrumentation functions ..."
How to install these functions on Heroku - on a server which is like a black box?
That message is referring to the adminpack module.
You can't install this on Heroku. This is appropriate because you don't control the database server, you control the database. adminpack provides functions for a superuser to modify the filesystem, which is a server-level consideration. Likewise, Heroku does not permit you to drop your database, create new users, or perform other superuser functions.
In exchange for taking your system-level control, Heroku performs all operations functions, such as applying OS- or database-level patches, setting up replication (fork/follow), backups, and migrating your database around planned or unplanned outages. If you'd rather do this yourself, you can of course run your own database in EC2 and point your Heroku application to that -- at which point you can install adminpack and administer to your heart's content.
All that said, you don't need adminpack to perform database-level operations. Your user has full control over your database, and PostgreSQL supplies everything you need for that out-of-the-box.