MySQL support not included in PHP - Wordpress Heroku
The CiviCRM installer checks for the presence of mysql_connect. Your Heroku environment has mysql_* extension disabled. The installer won't run if the environment doesn't pass CiviCRM's checks (as you've discovered).

This mysql_ library was deprecated in PHP5.5 and removed in PHP7. It looks like Heroku do offer mysql_ as an option you can enable through composer.json (see their docs on Heroku PHP extension support). 

Even so, it's likely easier to use a hosting environment known to support CiviCRM; if you're a CiviCRM pioneer on Heroku, you may have to deal with other features of that environment which CiviCRM hasn't been road-tested with. (Not a criticism of Heroku - but CiviCRM has some expectations of its host environment above and beyond the "average" PHP app.)

It's also possible that you could convert CiviCRM to use mysqli_* with a global replace of the mysql_* functions through the CiviCRM codebase ... if that works, the relevant ticket is CRM-16067: Replace deprecated mysql api with PDO or mysqli!

See also:


Undefined function mysql_connect on Heroku
Error during Drupal installation: MySQL support not included in PHP
php.net: mysql deprecation