It is not permissible for a father to give his zakaah to his son, because if the son is poor and the father is rich, the father is obliged to spend on his son, so if he gives him his zakaah, it is as if he is giving it to himself.
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (2/269): The obligatory charity (zakaah) should not be given to parents or to children. Ibn al-Mundhir said: The scholars are unanimously agreed that it is not permissible to give zakaah to one’s parents in a situation where the giver is required to spend on them, because giving zakaah to them means that they no longer need him to spend on them and that he is spared that duty, so he benefits from it, and it is as if he is giving it to himself.
Similarly, he should not give it to his son. Imam Ahmad said: Zakaah should not be given to one's parents or child, or son’s child, or grandfather or grandmother, or daughter’s child. End quote.
There are two exceptions:
1 – If the son is in debt, in which case it is permissible to give him zakaah, because the father is not obliged to pay off his son’s debt.
2 – If the father does not have enough to spend on his son.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said, in al-Ikhtiyaarat (p. 104): It is permissible to give zakaah to parents and other ascendants – meaning grandfathers and grandmothers – and to children and other descendents – meaning grandchildren – if they are poor and he is unable to spend on their maintenance, and also if they are in debt or are slaves who have a contract of manumission with their owners, or are wayfarers. If the mother is poor and has small children who have wealth, and spending on her will affect them, then she may be given some of their zakaah. End quote.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is it permissible for a person to give zakaah to his son?
He replied: That depends. If he wants to give it to him when he is obliged to spend on him, then that is not permitted. If he wants to pay off a debt on his behalf, such as if there has been a car accident and the car that he hit was damaged, and the car is valued at tens of thousands, then it is permissible for the father to pay it with his zakaah because of this accident. End quote from Majmoo’ Fataawa al-Shaykh Ibn ‘Uthaymeen (18/508). See also (18/415).
And Allaah knows best.