Why is it wrong for Desdemona to beg Othello to be friends with Cassio saying "for the love I bear Cassio"?
It is not morally wrong at all, but it is socially unacceptable in this sexist society. A woman, certainly a wife, is meant to be seen and not heard. Desdemona breeches her bounds as a wife because she crosses over into the military world of male reputation.
Desdemona is merely a victim of Iago's trap and Othello's jealousy. Desdemona is a bit naive, sure, and she has hurried into a marriage with a man whom she does not know well, but she is blameless in interceding for Cassio. "The love she bears Cassio," she believes, is Platonic, but it comes across--thanks to Iago's treachery--as romantic to Othello's jealous ears.
Ironically, Iago knows Othello better than Desdemona knows her husband. Iago knows that Othello is jealous of Cassio because he is white, young, good-looking, and Christian (the ideal European male). As an older, black, former slave and pagan, Othello has a deep-set inferiority complex. When Iago plants the idea that Desdemona and Cassio are having an affair, it flares his feelings of inferiority and jealousy. Othello, above all else, does not want to be made a cuckold: a man whose wife cheats on him.
Desdemona, playing the dutiful wife, wants to please her husband and his friends. She believes her duty is to cross over into the male military world and advocate for Cassio. Little does she know that her pleadings are leading to her and Cassio's demise.