My question is that if a Muslim man marries a Christian or Jewish woman and then he divorces her the third time is he allowed to take her back when she marries another Christian or Jewish man and he divorces her?

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 a man divorces his wife an irrevocable divorce and then she marries another man a serious marriage, and the latter divorces her or dies, the first husband is allowed to marry her again whether she is a Muslim woman or a woman from the People of the Book, and whether the second husband is a Muslim man or a man from the People of the Book according to the most preponderant view in our opinion here in Islamweb. Allaah Says (what means): {And if he has divorced her [for the third time], then she is not lawful to him afterward until [after] she marries a husband other than him. Then, if the other husband divorces her, it is no sin on both of them that they reunite, provided they feel that they can keep the limits ordained by Allaah.}[Quran 2:230].

Finally, it should be noted that it is better for a Muslim not to marry women from the People of the Book for many reasons which we already mentioned in Fataawa 84724 and 89418. Therefore, it is better for this man not to marry this woman again if there is no preponderant benefit, like fearing about his children when being separated from her, or her taking his children away from him and the like. 

Allaah Knows best.