Shaykh al-Islam Ibn Taymiyah was asked this question, and he replied: 

If the son is in debt and cannot pay it off, then he can take from his father’s zakaah, according to the more correct of the two views narrated in the madhhab of Ahmad and others. But if he is need of maintenance and his father does not have any money to spend on him, in this case there is a difference of opinion (among the scholars), but the more correct view is that it is permissible for him to take from his father’s zakaah. But if he is of independent means and does not need his father to spend on him, then he has no need of his zakaah. And Allaah knows best.