Gemara

24a - Mishnah: If a trap was set for Chayos, birds or fish before Yom Tov, one may not take from it on Yom Tov unless he knows that it was caught before Yom Tov;

A case occurred in which a Nochri brought fish to R. Gamliel. He said 'they are permitted, just I do not want to take from him'.

Gemara - Question: The case contradicts the law!

Answer: The Mishnah is abbreviated; it means as follows. Safek Muchan is forbidden; R. Gamliel permits. A case occurred...

Shmuel: The Halachah does not follow R. Gamliel.

According to other versions, Shmuel ruled like Tana'im in Beraisos who forbid Safek Muchan not involving Nochrim.

24b - Rav Papa: The Halachah is, if a Nochri brought a gift to a Yisrael on Yom Tov:

If there is attached produce of the same species, it is forbidden even after Yom Tov, bi'Chdei she'Ya'asu (until the time needed to pick them after Yom Tov);

If there is no attached produce of the same species within the Techum, it is permitted only if it came from within the Techum.

Something brought for a Yisrael is permitted for a different Yisrael.

Eruvin 40a: A turnip was brought to Rava's city (on Yom Tov); he saw that it was wilted. He permitted to buy it, for surely it was not picked today.

Question: Perhaps it was brought from outside the Techum!

Answer: Something brought for one Yisrael is permitted for another. All the more so this turnip is permitted, for it was brought for Nochrim (most Yisraelim do not buy on Yom Tov).

Rava (later) saw that many were brought. He forbade them, for surely they are brought for (all) Yisrael.

On Yom Tov Sheni, Nochrim detached myrtle for people that make crowns for Chasanim. Ravina permitted people to smell it immediately at night.

Rava bar Tachlifa: You should forbid, for people here are not Benei Torah (they might come to disgrace Yom Tov Sheni)!

Objection (Rav Shemayah): This implies that Benei Torah would be permitted. They should have to wait after dark bi'Chdei she'Ya'asu!

Rava: They must wait bi'Chdei she'Ya'asu.

Rishonim

The Rif and Rosh (3:2) bring versions of Shmuel's rulings, Rav Papa's Halachah, and the episodes in Eruvin.

Rif: If there is attached produce of the same species, it is forbidden even after Yom Tov, i.e. even to move it. Because one may not eat it, it is Muktzah.

Ran (DH ul'Erev): A Mishnah (Machshirin 2:6) says that if Yerek is sold on Shabbos in a city that is mostly Nochrim, one may buy it immediately, i.e. after Shabbos. If the city is at least half Yisraelim, one must wait until it could be brought from nearby. The Mishnah only discusses the time to bring it, for very little time is needed to detach.

Rambam (Hilchos Yom Tov 2:10): If a Nochri brought a gift to a Yisrael on Yom Tov:

If there is attached produce of the same species, or if he brought Chayos, birds or fish that could be caught the same day, it is forbidden until night, and he waits bi'Chdei she'Ya'asu;

If there is no attached produce of the same species, or if one can tell that the gift was detached or trapped before Yom Tov, it is permitted only if it was brought from within the Techum.

Something brought for a Yisrael from outside the Techum is permitted for a different Yisrael.

Rosh (ibid.): If there is attached produce of the same species, since he did not detach the produce before Yom Tov, this shows that he put them out of his mind. Even R. Shimon forbids such Muktzah; it is like dry figs and raisins. Rashi says that he must wait bi'Chdei she'Ya'asu to avoid benefit from Melachah done on Yom Tov. Yom Tov Sheni is more lenient, so we do not require waiting bi'Chdei she'Ya'asu after it. In Eruvin, Ravina wanted to permit because he understood that Rav Papa's Halachah applied only to Yom Tov Rishon. R. Gershom, the Rambam and Ra'avad agree.

Question (R. Tam, brought in Rosh): A Mishnah (Terumos 2:3) allows eating on Shabbos something cooked b'Shogeg on Shabbos. We are not concerned for benefit from Melachah of Shabbos. All the more so, we are not concerned for Melachah of a Nochri on Yom Tov!

Suggestion: Perhaps this is only regarding cooking, for the food could have been eaten raw.

Rejection: If one was Shochet b'Shogeg it is permitted as long as the animal was not Muktzah (e.g. there was a Choleh Bein ha'Shemashos)! R. Meir does not forbid Meleches Shabbos b'Shogeg on Shabbos. Even R. Yehudah agrees; he just makes a Kenas to forbid, lest one permit Mezid.

Answer (Rosh): Rashi holds that Chachamim forbid only what is common. It is not common for a Yisrael to cook on Shabbos b'Shogeg. R. Tam says that Chachamim decree to wait bi'Chdei she'Ya'asu lest a Yisrael tell a Nochri to do Melachah for him. What a Nochri does for himself or for Nochrim is permitted even that day. We cannot bring a proof for Rashi from the Hadasim, for perhaps Ravina does not decree.

R. Tam must say that bi'Chdei she'Ya'asu includes the time to bring it from the place he cut it, for otherwise there is still incentive to tell the Nochri on Yom Tov. Perhaps we are not so stringent, and it suffices to wait until it could be cut and brought from the closest place it is found. Rashi forbids bi'Chdei she'Ya'asu to all Yisrael. Presumably, R. Tam only forbids to the Yisrael for whom it was brought. Regarding something brought from outside the Techum the Gemara says that it is permitted to others (for whom it was not brought). The Gemara does not say so about something detached. Perhaps it did not bother, for bi'Chdei she'Ya'asu is a small difference (all must wait until Motza'ei Yom Tov, for it is Muktzah). According to R. Tam, one need not wait bi'Chdei she'Ya'asu if a Yisrael cooked on Shabbos.

Question (against R. Tam - Taz OC 515:1): Normally, Chachamim decree only if there is concern for an Isur mid'Oraisa. Where they decreed to protect a decree, the Gemara says that it was all one decree from the beginning, for otherwise the decree would not stand.

Question: Why does Rashi need to say that they are forbidden on Yom Tov due to Muktzah? In any case they are forbidden until bi'Chdei she'Ya'asu on Motza'ei Yom Tov, to avoid benefit from Meleches Yom Tov!

Answer #1 (Ba'al ha'Ma'or): If they were brought for one Yisrael, they are forbidden to another Yisrael only due to Muktzah.

Answer #2 (Ran DH Aval): If the Nochri picked them for himself, they are forbidden to Yisraelim only due to Muktzah.

Rosh (ibid.): The stringency of bi'Chdei she'Ya'asu is only when a Melachah mid'Oraisa was done. We are not concerned lest a Nochri took through Reshus ha'Rabim, for nowadays a Reshus ha'Rabim regularly frequented by 600,000 is not common. If Nochrim brought flutes for a eulogy we must wait bi'Chdei she'Ya'asu. We are more stringent there, for it becomes known. This is why one may never bury a Yisrael in a grave dug by Nochrim on Shabbos.

Poskim

Shulchan Aruch (515:1,5): Rambam (Hilchos Yom Tov 2:10): If a Nochri brought a gift to a Yisrael on Yom Tov:

If there is attached produce of the same species, or if he brought things that need to be trapped, it is forbidden to eat or move them that day, even for Yisraelim for whom they were not brought. They are forbidden at night until bi'Chdei she'Ya'asu.

Magen Avraham (2): The Yam Shel Shlomo says that even though one may not eat it on Yom Tov Sheni, one may move it then. This is wrong. We cannot learn from produce that left its Techum, for it may be eaten if it returns to its Techum.

Magen Avraham (3): The Gemara and Poskim discuss a gift. It seems that if the Yisrael hired the Nochri to bring before Yom Tov and the Nochri brought on Yom Tov, it is permitted at night immediately. If there is no concern that it was detached or trapped on Yom Tov, it is permitted on Yom Tov itself, for the Nochri did so for himself. The Mordechai says that in the episodes in Eruvin, a Yisrael hired the Nochri to bring before Yom Tov. Rava says that the Nochri did so for Nochrim; we must say that he means, for himself. If a Nochri lit a lamp for the sake of Yisraelim, it is forbidden for them, for he intended that they benefit from it on Shabbos. If he made shoes for them, they are permitted. It is not clear if food is forbidden like a lamp. Perhaps it is like shoes, for it is possible to benefit from it another day!

Mishnah Berurah (1) and Bi'ur Halachah (DH Doron): There is no difference whether the Nochri brought a gift, intended to sell to Yisraelim, or if a Yisrael hired the Nochri to bring before Yom Tov. This is clear from the Or Zaru'a, which is the Mordechai's source. However, when the Nochri was hired one may be lenient about things brought from outside the Techum, or even regarding Melachah if there is a Safek (whether or not it was done on Yom Tov).

Mishnah Berurah (2): If it is a Safek whether or not they were detached or trapped on Yom Tov; we are stringent. Some say that most people do not detach for the sake of the following day.

Mishnah Berurah (4): All benefit is forbidden, e.g. to smell Hadasim.

Kaf ha'Chayim (8): Muktzah does not apply to a Nochri's property regarding living animals, but it applies to attached produce or animals that need to be trapped.

Shulchan Aruch (ibid.): Bi'Chdei she'Ya'asu is the time for the Nochri to go to where he picked it, complete the Melachah and return to here. If one does not know from where he brought them, he allows time to bring them from outside the Techum.

Beis Yosef (DH u'Mah she'Chosav u'Perush and DH Aval): According to Rashi, bi'Chdei she'Ya'asu is the short time it takes to detach the produce. R. Tam includes the time to bring it. However, the Ran says that Rashi's opinion is primary, nevertheless he requires waiting until it could be brought from nearby. Hagahos Maimoniyos suggests that perhaps one must wait until the Melachah could be done during the day (for it is difficult to do it at night). He concludes that presumably we are lenient.

Magen Avraham (9): If the Nochri rode on a horse, we consider the time needed to bring in this way.

Kaf ha'Chayim (40): We are stringent about a Safek from where he brought them, even though the Isur is mid'Rabanan, because Yesh Lo Matirim.