Gemara

29a (Mishnah): Chachamim forbid eating cheese of Nochrim , but permit benefit from it.

Question (R. Yishmael): Why is it forbidden?

Answer #1 (R. Yehoshua): It is because they use stomachs of Neveilos to curdle it. (Note: in this Halachah, 'stomach' always refers to what is inside the stomach, unless we specify the skin of the stomach.)

Objection (R. Yishmael): The Isur of an Olah is more severe, yet the Torah permits a Kohen to suck the stomach of an Olah!

Answer #2 (R. Yehoshua): Rather, it is because they use stomachs of calves used for idolatry to curdle it.

Objection (R. Yishmael): If so, why may one benefit from it?!

35a - Answer #3 (R. Shimon ben Pazi): Perhaps a snake drank from (the milk, and poisoned) it.

R. Yehoshua didn't say so, for in Eretz Yisrael, they do not reveal the reason for a new decree for a year, lest someone not accept it.

Objection (R. Yirmeyah): If so, we should permit dry or aged cheese!

(R. Chanina): If something was exposed, and it dried or aged, it is permitted. If it contained venom, it would not dry or age properly.

Answer #4 (R. Chanina): There are always drops of milk in cheese.

Answer #5 (Shmuel): They curdle in the skin of stomachs of Neveilos.

Answer #6 (Rav Malkiya): They use pig lard to smooth the cheese.

Answer #7 (Rav Chisda): They use vinegar to curdle.

Answer #8 (Rav Nachman bar Yitzchak): They use sap of Orlah to curdle.

Question: According to Rav Chisda and Rav Nachman, it should be forbidden to benefit from cheese of Nochrim!

This is left difficult.

Rishonim

Rif (Chulin 42b): The Halachah is, one may not curdle with the skin of the stomach of a Neveilah. One may curdle with the stomach of an animal that a Nochri slaughtered, for the milk inside is like mere Peresh (a secretion).

Rosh (Chulin 8:52): This is the Rif's text. He explained similarly above, that we forbade a stomach cooked with the milk inside before R. Yehoshua retracted. We must also say that we forbade the stomach of a Kosher animal that nursed from a Tereifah before the retraction. The Gemara connotes otherwise. It seems that this was taught with a Tereifah that nursed from a Kesherah, and we permit both because the milk is gathered in its innards. This is not in Rashi's text. The words do not connote like the text of Rav Ashi (perhaps this should say Rashi - PF), but Seforim were emended to say like this. R. Tam says that according to ths text, we can distinguish. Here it says 'it is a mere Peresh', i.e. congealed milk in the stomach. The Mishnah that forbids a Kosher animal that nursed from a Tereifah and a stomach cooked with its milk discusses clear milk in the stomach. It is proper milk.

Rambam (Hilchos Ma'achalos Asuros 3:13): Letter of the law, cheese of Nochrim should be permitted, for Tamei milk cannot become cheese. Chachamim of the Mishnah forbade it, because they curdle with skin of stomachs of animals that they slaughter, which are Neveilos. The skin is very small compared to the milk. It is not Batel because it is Ma'amid (curdles), therefore, all the cheese is forbidden.

Ra'avad: The Gemara gave (also) other reasons to forbid, e.g. due to drops of Tamei milk in the cheese, or they smooth the cheese with pig lard. Do not say that the concern is only for dry cheese. Just the opposite (there is more concern for wet cheese)! I heard that the Ge'onim who forbade cheese of Nochrim curdled with Heter (see the next Halachah in the Rambam) did so according to letter of the law. It was not a mere decree.

Rambam (14): Cheese that Nochrim curdle through herbs or fruit juice, e.g. fig sap, and one tastes them in the cheese, some Ge'onim forbid it, because Chachamim already forbade all cheese of Nochrim, whether it was curdled with Isur or Heter. We forbid the latter due to what was curdled with Isur.

Beis Yosef (YD 115 DH v'Chasuv b'Hagahos): Since R. Tam rules like R. Yehoshua ben Levi, we should forbid cheese of Nochrim even in places where they curdle with flowers. The Rashba says that where it is known that they do not mix Tamei milk with Tahor, for they despise Tamei milk, and where we know that they do not use pig lard to smooth the cheese, for they eat cheese on their fast days when they may not eat meat, and we know that they do not curdle with skin of stomachs of Neveilos, rather, with flowers, even so one may not eat cheese of Nochrim, for it was forbidden with a vote, like their wine. Even though the reason no longer applies, the Isur remains.

Rashba (Teshuvah 5:56): Rav Nachman must hold that Chachamim were not stringent to forbid benefit from cheese of Nochrim, for it is not Vadai that they use Orlah.

Note: The Rashba was discussing Orlah. Surely we could answer similarly for Rav Chisda. Presumably, the Gemara did not give the Rashba's answer for them because it did not accept their reasoning (Tosfos ha'Rosh Shabbos Reish 51a says similarly). E.g. one cannot taste the Orlah, so we forbid eating only due to Hana'ah, therefore, it is unreasonable to forbid eating and permit other Hana'ah.

Poskim

Shulchan Aruch (YD 115:2): Cheese of Nochrim is forbidden because they curdle in the skin of the stomach of animals that they slaughter, which are Neveilos.

Beis Yosef (DH Gevinos): The Gemara gave many reasons why cheese of Nochrim was forbidden. The Tur omitted the reasons of Rav Nachman and Rav Chisda, for the Gemara left them difficult, and R. Yehoshua ben Levi's reason, for (nowadays) we are not concerned for Giluy. Tosfos says in the name of R. Tam that there is not a simple reason to forbid nowadays, for we hold like R. Yehoshua ben Levi against even R. Yochanan (and all the more so, against other Amora'im). There is no concern lest he mixed in Tamei milk. Nochrim are not so foolish; Tamei milk does not curdle. The only reason is Giluy, and snakes are not common among us. One need not be concerned lest he curdle in a Neveilah stomach, for this is a Sefek Sefekah (perhaps he did not, and even if he did, perhaps this is permitted).

Taz (9): Why isn't it Asur due to meat and milk? Mid'Oraisa, meat and milk does not apply, for it is cold. Even though it curdles, this forbids only mid'Rabanan. The Ri mi'Gash says that the meat is less than one part in 60 and cannot be tasted, so it is Batel, but a Ma'amid is never Batel. The Ra'avad said that whenever Chachamim forbade something of Nochrim, Bitul in the majority does not help, lest people breach it, since it is common thatit has taste. The Beis Yosef explained that the Rambam holds that the stringency of cheese is due to Ma'amid. Regarding butter, he needed to say that drops of (possibly Tamei) Kum are not Batel because they are not mixed. The Rashba says that they are mixed, but Bitul does not apply to what Chachamim forbade of Nochrim, like the Ra'avad said.

Shulchan Aruch (ibid): Even if they curdle with grass, it is forbidden.

Taz (10): This is because the Mishnah explicitly forbids cheese, and Chachamim did not distinguish, for even nowadays some curdle in skin of (stomachs of) Neveilos.

Gra (13): The Gemara asked 'if so, dry or aged cheese should be permitted!' This shows that it was clear that all cheese of Nochrim is forbidden without distinction, and we asked why this is so. The Gemara answered that it was a decree, i.e. it was forbidden through a vote.