Changing MySQL password via WHM – does this affect WordPress?

If that changes the MySQL password that WordPress is using, and it sounds like it does, then yes you will have a problem, but not a big one. You’d need to edit wp-config.php and change define('DB_PASSWORD','oldpass'); to define('DB_PASSWORD','newpass'); Beyond that, the question is a CPanel question and would be off topic here.

http://codex.wordpress.org/Editing_wp-config.php#Set_Database_Password