"He is teaching me Chinese."
Translation:O bana Çince öğretiyor.
(Native Turkish speakers, correct me if I'm wrong)
Beni would make you sound like you're saying "He is teaching me" where "me" is some subject that is being taught in class.
He is teaching Chinese. Chinese is the direct object, it's the subject of study that is being taught. "Me" is not being taught.
Bana means "to me". It is the indirect object (it is not the subject of study being taught).
In essence, this sentence actually translates to
He is teaching Chinese to me.
because me or bana in those sentences is an indirect object and this requires dative in Turkish. The direct object is "çince"
you can also think of it as "to teach something to someone": "birine bir şey(i) öğretmek"