Is it considered a sin for girls and boys to get to know each other or exchange songs via private messages in a forum I administer, and should I give up its administration?

If the forum is regulated by Islamic strictures, then there is no harm in what some members do secretly from impermissible acts. It is your duty to advise and guide the members, and to warn them against what contradicts Islamic law. If an objectionable act appears in the forum, it is obligatory to denounce it, remove it, and advise its perpetrator. If you are unable to regulate the forum according to Islamic strictures, then it is obligatory to close it.