“Μου” takes tonos only when used as an indirect object and there is a need to distinguish it from the possessive pronoun.
Ο καθηγητής μού εξήγησε το μάθημα. (The teacher explained the lesson to me)
Ο καθηγητής μου εξήγησε το μάθημα. (My teacher explained the lesson)
Ο καθηγητής μου μου εξήγησε το μάθημα. (My teacher explained the lesson to me)
In the first example we need to put a tonos on “μου”, to show that it is indirect object and not a possessive pronoun (second example).
In the third example we have two “μου”. It is clear that the first one is a possessive pronoun and the second one is indirect object. Therefore there is no need to put a tonos on the second “μου”.
The same is true with the corresponding personal pronouns of the other persons “σου”, “του”, “σας” etc.