Does a husband have to take care of children in a marriage who are not his?
When you marry a woman who has children from a prior marriage, you are accepting responsibility for those children. "But if anyone does not provide for his own, and especially for those of his household, he has denied the faith and is worse than an unbeliever" (I Timothy 5:8). The use of the word "household" is pertinent to this discussion. It doesn't refer to just a person's biological children. It includes all people living with him, including the servants.
We also find people raising the children of others in the Bible. "And Mordecai had brought up Hadassah, that is, Esther, his uncle's daughter, for she had neither father nor mother" (Esther 2:7). It is a part of pure religion. "Pure and undefiled religion before God and the Father is this: to visit orphans and widows in their trouble, and to keep oneself unspotted from the world" (James 1:27). "Visit" means more than stopping by for tea. It means to see to their welfare. To claim it is not your responsibility is to deny what God has said.