Gemara

A deer was trapped (by Nochrim) in the Reish Galusa's house on Yom Tov Rishon, and it was slaughtered on Yom Tov Sheni. Rav Nachman and Rav Chisda ate from it, but Rav Sheshes did not.

Rav Nachman: Rav Sheshes does not wish to eat deer meat! (There is no reason to forbid. Yesterday was Chol, or today is.)

(Rabah bar Shmuel - Beraisa): R. Yosi agrees about the two days of Yom Tov of Chutz la'Aretz.

After Rav Sheshes heard this Beraisa, he retracted.

Rav Ashi: The deer was not trapped on Yom Tov, rather, it was brought from outside the Techum. Those who ate held that something brought for one Yisrael is permitted for another;

Rav Sheshes did not eat, for he holds that everything brought to the Reish Galusa's house is brought for Chachamim. Really, he never retracted.

Shabbos 122a (Mishnah): If a bathhouse is open on Shabbos, if the majority of the Chazakah are Nochrim, one may bathe in the bathhouse immediately (on Motzei Shabbos). If the majority are Yisraelim, one must wait after Shabbos bi'Chdei she'Ya'asu (the time to do the Melachah, i.e. heat the water).

151a (Mishnah): If they made a coffin or grave [for a Nochri or to sell, a Yisrael may be buried in it].

Question: We should have to wait bi'Chdei she'Ya'asu!

Answer (Ula): The case is, the grave was dug on a major road. (Yisraelim do not bury there. Surely it was for the sake of Nochrim.)

Beitzah 24b (Rav Papa): The Halachah is, if a Nochri brought Peros on Yom Tov, if there are attached Peros of that species, we forbid the Peros until the time to pick them after Yom Tov.

Rishonim

Rambam (Hilchos Shabbos 6:23): If a Yisrael did Melachah on Shabbos b'Mezid, he may never benefit from it. Other Yisraelim may benefit from it on Motza'ei Shabbos immediately. If a Yisrael cooked on Shabbos b'Mezid, others may eat it after Shabbos, but he may never eat it. If he was Shogeg, he or others may eat it immediately after Shabbos. The same applies to everything similar.

Magid Mishneh: The Ramban forbids to everyone bi'Chdei she'Ya'asu, even if the Melachah was b'Shogeg, lest they benefit from Ma'aseh Shabbos.

Question: If a Nochri prepared needs of burial on Shabbos, the Ramban (cited in Magid Mishneh 6:5) says that no one may use them bi'Chdei she'Ya'asu because we make a uniform decree. Why doesn't the Rambam forbid here to everyone bi'Chdei she'Ya'asu?

Answer (Lechem Mishneh): In any case the decree will not be uniform, since we will not forbid them permanently like we forbid the Mevashel, therefore we are not stringent about others at all.

Rosh (Chulin 1:19): We are stringent to require waiting bi'Chdei she'Ya'asu when a Nochri did Melachah for a Yisrael, or if a Yisrael caused Melachah to be done on Shabbos in a forbidden way, e.g. he left a pot of raw food on the fire just before Shabbos. This stringency does not apply when a Yisrael did Melachah on Shabbos. This is uncommon, so we do not decree.

Rosh (Beitzah 3:2): If a Nochri brought a gift to Yisrael, if some of that species is attached, it is forbidden at night bi'Chdei she'Ya'asu (until the time to detach it). Even R. Shimon admits regarding such Muktzeh. Since he did not detach it before Yom Tov, it is like dry figs and raisins. Rashi explains that it is forbidden at night bi'Chdei she'Ya'asu so he will not benefit from Melachah of Yom Tov. After the first night, even though it is Yom Tov (Sheni), it is permitted in any case. If the second day is Chol, he waited the proper amount of time. If the second day is Kodesh, they were picked on a Yom Chol. Rashi brought a proof from an egg laid on Yom Tov Rishon. It is permitted on Yom Tov Sheni. Also, a deer was trapped on Yom Tov Rishon, and it was permitted on Yom Tov Sheni (Eruvin 39b). Rava permits to place an Eruv Tavshilin on Yom Tov Rishon for Yom Tov Sheni and stipulate. Also, Rashi proved that Rav Papa's law of waiting bi'Chdei she'Ya'asu was said primarily regarding Yom Tov Rishon. Once, Nochrim cut myrtle for Yisre'elim on Yom Tov Sheni, and Ravina permitted on Motza'ei Yom Tov immediately (Eruvin 40a). He must hold that the stringency of bi'Chdei she'Ya'asu was said regarding Yom Tov Rishon, because (our calendar is fixed, so) we know that Yom Tov Rishon is really Yom Tov. We keep two days merely to preserve the custom of our ancestors. Also R. Gershom, R. Yakov ben Yakar, R. Klonimus, the Rambam and Ra'avad do not require bi'Chdei she'Ya'asu after Yom Tov Sheni.

Rosh: R. Tam challenged Rashi, who said that bi'Chdei she'Ya'asu is to avoid benefit from Meleches Yom Tov. If one cooked on Shabbos b'Shogeg, he may eat it if he was sick (when Shabbos began) and healed, so the food was not Muktzeh (Chulin 15b). He benefits from Meleches Shabbos! Also, R. Meir holds that there is no Isur of Meleches Shabbos if he was Shogeg. Even R. Yehudah forbids because he decrees Shogeg due to Mezid (Gitin 53b), but not due to benefit from Meleches Shabbos b'Shogeg. All the more so, one may benefit from what a Nochri did on his own! We can defend Rashi and say that we forbid more when a Nochri did for a Yisrael than when a Yisrael did b'Shogeg, for Chachamim did not decree about unusual matters.

Rosh (ibid.): R. Tam explains that it is a decree to wait bi'Chdei she'Ya'asu, lest one tell a Nochri to bring on Yom Tov in order to eat it immediately at night. This is why one must wait after Yom Tov Rishon. We decree only if the Nochri did so for the Yisrael. If he did so for himself, it is permitted immediately at night. We learn from a city of mostly Nochrim. One may bathe in a bathhouse immediately after Shabbos (Shabbos 151a). Even on Shabbos one may benefit from what a Nochri does for himself. If he drew water for his animals, a Yisrael may use it, and similarly if he lit a lamp or made a ramp for himself (Shabbos 122a). If something was trapped by itself, this decree does not apply. There is no proof from the deer, for it was trapped by itself. We cannot learn from Ravina. Perhaps he does not decree bi'Chdei she'Ya'asu on Yom Tov Rishon or Sheni. Even had they cut on Yom Tov Rishon, he would have permitted immediately [at night]! The Gemara needed to specify that it was Yom Tov Sheni (and did not say Yom Tov Stam) to teach that even on Yom Tov Sheni, Rav Tachlifa wanted to forbid because they are not Bnei Torah.

Rosh (ibid.): According to R. Tam, he should have to wait the time for the Nochri to go to where he trapped the deer or fish or picked the Peros, and return. If he must wait only the time to trap or pick, there is still concern lest one tell a Nochri to do so in order to eat it shortly after Yom Tov! Perhaps Chachamim were not so stringent. It seems that it suffices to wait until he could have brought them from a place where they are found near the city. He need not wait the time to bring them from where the Nochri actually brought them. Bahag wrote like R. Tam, and said that if two days of Yom Tov preceded Shabbos, one must wait on Motza'ei Shabbos bi'Chdei she'Ya'asu, lest he tell the Nochri on Yom Tov for the sake of Shabbos. R. Yechiel permits on Shabbos. One need not be so stringent; this is reasonable. According to Rashi, any Yisrael must wait bi'Chdei she'Ya'asu, lest he benefit from Meleches Shabbos. According to R. Tam, the decree is lest he command a Nochri, so presumably it is permitted to other Yisre'elim immediately. We are not concerned lest Reuven command for the sake of Shimon. One does not sin for the sake of others. Here the Gemara did not say so, like it said about a gift brought from outside the Techum. Since it is forbidden to all Yisrael on Shabbos, the Tana did not elaborate to teach about the short time after Shabbos. According to R. Tam, if one cooked on Shabbos, one need not wait after Shabbos, for there is no need to decree.

Rosh: If the species is not available attached, within the Techum it is permitted, and outside the Techum it is forbidden, but at night one need not wait bi'Chdei she'Ya'asu. Chachamim made this stringency only if an Isur Torah was done. When it is available attached, we forbid not only eating, rather, even to move it. Since one may not eat it, it is Muktzeh.

Rashi (Chulin 15a): No one may benefit from Meleches Shabbos, i.e. what was done b'Mezid. Therefore, all agree that no one may benefit until bi'Chdei she'Ya'asu.

Poskim

Shulchan Aruch (OC 318:1): If one cooked on Shabbos (Rema - or did any other Melachah) b'Mezid, he may never eat it. Others may eat it immediately after Shabbos.

Bach (DH u'Mah she'Chosav): The Poskim require bi'Chdei she'Ya'asu only when a Nochri did the Melachah, lest Yisraelim come to ask Nochrim to do Melachah. Rashi says that it is to avoid benefit from Meleches Shabbos. This applies even when a Yisrael transgressed.

Rebuttal (Magen Avraham 2): Rashi would say that we did not decree about this, for it is uncommon (like the Rosh said).

Shulchan Aruch (ibid.): If he was Shogeg, it is forbidden on Shabbos also to others. After Shabbos it is permitted also to him immediately.

Beis Yosef (DH v'Chosav): The Rashba, Ran and Magid Mishneh say that if someone cooked or uprooted food for a Choleh, all permit it immediately on Motza'ei Shabbos. Bi'Chdei she'Ya'asu is needed only when there was Chilul Shabbos b'Shogeg or b'Mezid. Here, the Melachah was permitted. It is forbidden to a healthy person only due to Muktzeh, or due to a decree lest one do extra for a healthy person, so we do not require bi'Chdei she'Ya'asu.

Rema: Even if he cooked through a Nochri, it is forbidden on Shabbos.

Magen Avraham (7): It is permitted immediately on Motza'ei Shabbos. Since the Nochri cooked b'Heter, one need not wait after Shabbos the time to cook.

Rema (2): If one detached fruits on Shabbos for a Choleh, even if he was sick from before Shabbos, it is forbidden to a Bari on Shabbos, for it constantly grows on Shabbos, so it is Muktzeh.

Kaf ha'Chayim (35): One must wait after Shabbos the time to do a Melachah (that was done for a Choleh), but not if the Isur was due only to Muktzeh or a decree lest people do extra.