The Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1835-1909) rules that one may not drink milk produced by a gentile on Pesach, given the possibility that the cow may have been fed Hametz. Drinking the milk that a cow produces from the Hametz it was fed amounts to deriving Hana'a (benefit) from Hametz, which is, of course, forbidden on Pesach.
Chacham Ovadia Yosef in; Halichot Olam, Helek 1, page 305, however, disagrees with this ruling of the Ben Ish Chai. He cites the Beit Efrayim who addresses this very question and allows drinking milk produced by a gentile during Pesach, for three reasons. Firstly, the Hametz belongs to the gentile when it is fed to the cow, and it is perfectly permissible for a gentile to feed his animal Hametz on Pesach. Secondly, the animal eats many different kinds of foods, and not only Hametz, and so the milk is not necessarily produced specifically from Hametz. Finally, it is uncertain altogether with the milk one purchases was produced by a cow that was fed Hametz. Therefore, Chacham Ovadia rules that one may, in fact, drink milk produced by a gentile on Pesach. Nevertheless, some people abide by the stringency of the Ben Ish Chai, and make a point of purchasing all their milk before Pesach. Strictly speaking, however, this is not necessary.
It should be noted that this discussion does not involve the separate issue of Chalav Yisrael whether it is forbidden to drink milk generally if was not produced by a Jew, which is a different topic entirely.
In conclusion, then, in situations where it is permissible in general to drink milk produced by gentiles, one may do so even on Pesach, and even if the gentile produced the milk on Pesach. Nevertheless, there are those who are stringent in this regard and purchase all their milk for Pesach before the holiday.
