All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If the children are young or are old but are unable to gain lawful money, then they can take from their father's wealth even if it is gained by forbidden means. The ruling that applies to these dependents is the ruling of the person who is obliged to eat the meat of a dead animal in order to repel the harm from him. However, when they do not need the money of their father which is gained from forbidden resources, then it is forbidden for them to consume the forbidden money, because when they acquire lawful money, they are no longer in a state of necessity. Allaah has permitted us to consume what is forbidden only for a necessity. Allaah says (interpretation of meaning): {But if one is forced by necessity without willful disobedience nor transgressing due limits, then there is no sin on him. Truly, Allaah is Oft-Forgiving, Most Merciful.}[2:173]. However, when the state of necessity ends, the matter returns to its original state, which is that it is prohibited. So the children are sinful in consuming unlawful money if they are able to gain lawful money and are not in a state of inability or necessity. Therefore, the money is forbidden on the children, and they will be accountable for it, and it will not be blessed. The Prophet said: "The flesh that grows from ill-gotten money, the fire is more appropriate for it." [At-Tirmithi]
As regards what the questioner ascribed to Shaykh Ibn Al-'Uthaymeen that what is gained through forbidden sources is not forbidden for whoever acquires this money through a permissible way, this needs details. If it was meant that whoever steals something for instance or takes it by force, and then gives it as a gift or sells it to another person who knows that it is stolen or obtained by force, then it is not permissible for the latter to buy it or accept it as a gift, while knowing that it had been stolen or taken by force. If he accepts it as a gift or buys it, he is sinful, and it cannot be said that he has taken it in a lawful manner, as this is not what Shaykh Ibn Al-'Uthaymeen meant. But if he takes it not knowing that it had been stolen then there is no harm on him, and this is correct.
Allaah knows best.