Gemara

(Beraisa): If Leah needed to bring birds due to a birth or Zivah, she may leave money in the box, immerse, and eat Kodshim at night.

Eruvin 32a - Suggestion: She relies on Chazakah Shali'ach Oseh Shelichuso! (We assume that a Shali'ach fulfills his mission, i.e. the Kohen bought birds and offered them.)

Rejection (Rav Nachman): No, she relies on Rav Shemayah's teaching;

Pesachim 90a (Rav Shemayah): There is a Chazakah that the Beis Din of Kohanim does not leave [the Mikdash] until they finish [offering birds for] all the money in the box.

(Rav): We slaughter Korban Pesach and throw [the blood] for a Tevul Yom or a Mechusar Kipurim, but not for one who is Tamei Sheretz;

(Ula): We slaughter and throw even for one who is Tamei Sheretz.

Rav is more stringent about a Tamei Sheretz, for he must do an action [to become permitted]. A Tevul Yom becomes permitted automatically [at dark].

Question: Rav permits for a Mechusar Kipurim, even though he must bring birds!

Answer #1: The case is, he has the birds in his hand [he is ready to offer them].

Question: If so, we should slaughter for a Tamei Sheretz right in front of a Mikveh!

Answer #1: We are concerned lest he not immerse.

Objection: If so, we should also be concerned lest the Mechusar Kipurim not bring his Kaparah!

Answer #2 (to Question (h)): Rather, the case is, he gave the money for the birds to Beis Din. This is like Rav Shemayah taught [above].

Eruvin 31a (Rav Yechi'el): Chazakah Shali'ach Oseh Shelichuso.

(Rav Nachman): We rely on the Chazakah only for mid'Rabanan laws.

(Rav Sheshes): It applies even for Torah laws.

Support (Rav Nachman, for himself - Mishnah): People far from the Mikdash may eat Chodosh (grain that took root after last Pesach) after midday [of Nisan 16, the day we offer the Omer].

[This is] because Beis Din is not lazy. (They would not delay past midday.)

Inference: Beis Din is not lazy, but a regular Shali'ach could be lazy! (The Chazakah would not apply)!

Rejection (Rav Sheshes): Surely, Beis Din offered it by midday. For a regular Shali'ach, Chazakah teaches only that he fulfils his mission and offered it sometime [when it may be offered, i.e.] during the day.

Chulin 12a - Question (Rav Dimi bar Yosef): If Reuven asked a Shali'ach to tithe for him and later found the Ma'aseros separated, what is the law?

Answer (Rav Nachman): He may not assume that they were separated properly.

If Ploni overheard and tithed, it is invalid, for one cannot tithe without the owner's request.

Gitin 64a: We do not rely on Chazakah Shali'ach Oseh Shelichuso to be lenient, only to be stringent.

Rishonim

Rambam (Hilchos Terumos 4:6): If Reuven told Shimon 'take Terumah for me', and Shimon went, and Reuven found the pile tithed, there is no Chazakah that it is tithed. For Isurim, we do not say Chazakah Shali'ach Oseh Shelichuso to be lenient, only to be stringent. Perhaps someone else tithed without permission.

Rosh (Eruvin 3:3): R. Shimshon ha'Zaken rules like Rav Sheshes, for Ge'onim say that the Halachah follows him in Isurim, and Rav Nachman in monetary laws [when they argue with each other]. R. Tam rules like Rav Nachman. The Chazakah Shali'ach Oseh Shelichuso is only to be stringent (Gitin 64a). Chulin 12a concludes that Ein Chazakah Shali'ach Oseh Shelichuso. Rav Shemayah answers for Rav Nachman that there is a Chazakah for the Beis Din of Kohanim. This shows that he holds like Rav Nachman.

Rebuttal (Rosh): Rav Sheshes says Chazakah Shali'ach Oseh Shelichuso only when the Shali'ach must be concerned that if he does not fulfill his mission, the Meshale'ach will rely that he fulfilled it, and transgress. Therefore, the Shali'ach is not concerned if he does not tithe. The Meshale'ach will know that the pile was not tithed if he does not see Terumah near the pile. Perhaps Rav Shemayah does not teach why the woman relies on Kohanim, rather, why Rav says that we do not slaughter and throw for a Tamei Sheretz, even though there is a Mikveh available, for an action must be done. We are concerned for negligence, lest he not immerse [today, and be forbidden to eat Kodshim at night]. We do slaughter and throw for a Mechusar Kipurim. Even though an action must be done, we are not concerned, due to Rav Shemayah's Chazakah. Since R. Tam's proofs are rejected, we need not contradict the Ge'onim's rule.

Note: The Ri (in Tosfos 32a DH Rav) rejected R. Tam's proofs like the Rosh did.

Poskim

Shulchan Aruch (OC 409:8): If Reuven told Shimon to receive an Eruv from even an animal, and Reuven saw that Shimon received it, it is a valid Eruv, even if he did not see Shimon place it. Chazakah Shali'ach Oseh Shelichuso.

Magen Avraham (409:17): The Chazakah is even for an Isur Torah, e.g. if the Meshale'ach will transgress if the Shali'ach does not fulfill his mission, like here. However, if one told a Shali'ach to tithe, and he found the Peros tithed, there is no Chazakah that [his Shali'ach] tithed. The Shali'ach is not concerned to tithe, for if the Meshale'ach will not see the Terumah near the pile, he will know that it was not tithed. Therefore, we are concerned lest the Shali'ach did not tithe, and someone else tithed. This is difficult, for in YD 331 (below), if one found the Peros tithed, we say that the Shali'ach tithed, unlike Rav Nachman! This requires investigation.

Shulchan Aruch (YD 331:34): If Reuven told Shimon 'take Terumah for me', and Shimon went, and Reuven does not know whether or not he was Torem, and he found the pile tithed, there is no Chazakah that it is tithed. Perhaps someone else was Torem without permission.

Taz (19): The Rambam and R. Tam apply the Chazakah only to be stringent. The latter opinion applies it even to be lenient, but only if he knows that his pile was tithed. We are not lenient if it is a Safek.

Rema: Some say that in such a case the Chazakah is that it is tithed, for the Chazakah Shali'ach Oseh Shelichuso even for an Isur Torah. However, if he does not know whether or not the pile was tithed, he may not eat from it, for Shimon thinks 'surely Reuven will not touch [eat from] it until he knows that it was tithed.'

Rebuttal (Gra 80): In Chulin, we say that if he found the tithes separated, he may not assume that they were separated properly!

Mishneh l'Melech (Hilchos Bechoros 4:1): A man made a Shali'ach to sell part of his animal to a Nochri to exempt it from Bechorah. For Torah laws, we say Chazakah Shali'ach Oseh Shelichuso only to be stringent, like it says in Gitin. However, sometimes we rely on it to be lenient. The Ri explains that for Terumah, the Shali'ach knows that Reuven knows how much Tevel there was, so if he sees the full amount, he will know that it was not tithed. If he did not know, Chazakah Shali'ach Oseh Shelichuso, for he fears lest Reuven rely on him. The Rosh says that if Reuven does not see the tithes next to the pile, he will not rely on him. The Shali'ach must leave them, for the owner has Tovas Hana'ah (decides to whom to give them). One could explain the Ri like the Rosh. Or, the Ri explains that in Chulin, Reuven told Shimon to give the Terumah to a Kohen, or to keep it if he himself is a Kohen. Reuven will assume that it is tithed only if he knew the measure. The Rosh holds that Reuven kept the Tovas Hana'ah. He did not say that Shimon reasons 'no one else heard...' Perhaps he holds that when Reuven gave Tovas Hana'ah to Shimon, the Chazakah applies, for Shimon knows that others have incentive to be Torem! The Ri agrees to the Rosh's law. If he gave Tovas Hana'ah, and he does not know the amount, surely the Chazakah applies, like regarding Eruv. However, we need not make an argument. Presumably, the Ri and Rosh argue only about the case in Chulin, but all apply the Chazakah whenever the Meshale'ach will rely on the Shali'ach.

Mishneh l'Melech: Perhaps the Heter applies only when the Meshale'ach might transgress immediately, but not regarding Bechorah. The Shali'ach is lazy, for assumes that he has more time before it will gave birth. [Also, perhaps the calf will be a female, which in any case gets no Kedushah! - PF] Even Rav Sheshes applies the Chazakah only from the end of the day. He holds that Beis Din of Kohanim are more zealous than a Stam Shali'ach, and they fulfill the mission by midday.

Mishneh l'Melech (DH Ho'il): Rav holds that we do not slaughter and throw if we must rely on the owner to immerse. All the more so, we do not rely on his Shali'ach [to offer his Kaparah]! Only the Meshale'ach may rely on this. Rav Shemayah explained that we rely on Beis Din, which is a strong Chazakah. This was the Ri's intent. Semag (Aseh 48) says so. The Rif, Rambam, R. Tam and Rashba do not rely on the Chazakah to be lenient for a Torah Isur. We must be concerned for their opinion.

R. Akiva Eiger: At the very end, the Mishneh l'Melech says that perhaps even if someone else (not a Shali'ach) took Ma'aser, it is valid if all the Peros are the same quality. (There is no reason for the owner to insist that his Shali'ach do it, since everyone must give exactly a 10th.)