I'm not sure what Heroku gives you as a backup file, but assuming it was created with the mysqldump tool (for MySQL) o rpg_dump tool (for PostgreSQL), you'll be able to log into your Forge server (over ssh) and use the mysql or pg_restore commands to add it to a database within MySQL or PostgreSQL respectively.
All of which means you'll need to copy the backup file to your server, log into your server, and run one of the above commands to get the copied database into your MySQL or PostgreSQL.