Gemara

48a (R. Chanina of Tirasa - Beraisa): If four lambs were slaughtered for Shtei ha'Lechem, we throw the blood of two lambs Lo Lishmah (to permit Haktaras Eimurim and eating the meat.)

Question (R. Yochanan): Do we transgress (Zerikah Lo Lishmah) in order to gain (salvage the other lambs)?!

If limbs of a Chatas were mixed with limbs of an Olah, Chachamim do not permit to burn all of them, in order to gain (burn the limbs of the Olah)!

Answer: We transgress in a Chatas in order to gain in a Chatas (the same matter). We do not transgress in a Chatas in order to gain in an Olah.

80a (R. Chiya): If a Todah became mixed with its Temurah and one of them died, there is no way to offer the remaining animal

We cannot offer it with bread, perhaps it is the Temurah. We cannot offer it without bread, perhaps it is the Todah!

Question #1: Why can't he offer the remaining animal with bread, and stipulate 'if this is the Todah, this is its bread. If not, the bread is Chulin'?

Answer (Rebbi): One may not bring Chulin into the Azarah.

Question #2 (Levi): Why can't he bring another animal with bread, and stipulate:

If the remaining animal is the Temurah, this is a Todah and its bread. If the remaining animal is the Todah, the new animal is Mosar Todah!

Answer (Rebbi): We do not bring Mosar l'Chatchilah!

Pesachim 59a: A Mechusar Kipurim must bring a Korban before eating Kodshim. If he did not do so before the afternoon Tamid on Erev Pesach, to enable him to fulfill Korban Pesach (which has Kares for one who neglects it), we override the Aseh of Hashlamah (not to offer after the afternoon Tamid), which does not have Kares.

88b (Beraisa): If the skins of the Pesachim of five groups became mixed, and a wart was found on one (we do not know which Korban was Pasul), they are exempt from Pesach Sheni.

Question: Each could bring an animal and stipulate 'if ours had the Mum, this is Pesach Sheni. If not, this is a Shelamim'!

Answer #1: We cannot, due to Chazah v'Shok. (If it is a Shelamim, a Kohen must eat the chest and foreleg). Alternatively this reduces the time to eat Shelamim.

Answer #2: We cannot, for Dam Pesach is poured on the Mizbe'ach, and only once. Dam Shelamim must be thrown on two opposite corners.

Even though one Zerikah suffices, and even if it was poured, that is only b'Di'eved. It is not l'Chatchilah.

Kidushin 55a (Mishnah): If a male animal was found near Yerushalayim, it is an Olah.

(R. Oshaya): The case is, the finder accepted on himself to bring the Korban that the loser needed to bring. The Mishnah is like R. Meir, who says that one can profrane Kedushas ha'Guf (something used for Avodah in the Mikdash) b'Mezid.

55b Objection (R. Yochanan): Should one transgress (profane the animal) to help the one who lost it?!

Shabbos 4a - Question (Rav Bivi bar Abaye): If Reuven stuck a dough in an oven on Shabbos, may he remove it before it bakes, to avoid a Chiyuv Chatas?

(Rav Shila): He was Shogeg. The question is whether someone else may remove it to exempt him.

Objection: One may not sin in order to save another from sin!

Eruvin 32a (Beraisa - Rebbi): If a Chaver (someone trustworthy about Ma'aser) said 'go fill this basket with dates from my tree', Shimon may eat without tithing. (Surely, the Chaver took Terumah.)

R. Shimon ben Gamliel says, he may not eat until tithing Vadai. A Chaver would not separate Terumah Lo Min ha'Mukaf (far from the Peros it exempts).

32b: Rebbi holds that it is better for a Chaver to transgress a minor matter (Lo Min ha'Mukaf), lest an Am ha'Aretz transgress a severe matter (eating Tevel). R. Shimon ben Gamliel disagrees.

Chalah 2:3 (Mishnah): If one cannot make a dough in Taharah, he makes less than a Kav at a time (to exempt it from Chalah). He does not make a big dough (and separate Chalah) b'Tum'ah;

R. Akiva says, he makes a big dough b'Tum'ah. He does not make a Kav at a time. Just like he separates Tahor Chalah (from a Tahor dough), he separates Tamei Chalah. This way, Hash-m has a share in it, but not if he makes small doughs.

Tosefta (Chalah 1:9): Chachamim said to R. Akiva 'we do not tell someone to sin in order to gain, or to ruin in order to fix.'

Rishonim

Rambam (Hilchos Ma'aser 10:10): Even though a Chaver is not suspected to be Torem Lo Min ha'Mukaf, in order to save an Am ha'Aretz from transgressing, he does so.

Rambam (Hilchos Bikurim 8:11): If one was further than four Mil (a Mil is about a kilometer) from water, he may make a dough in Taharah and and separate Tamei Chalah.

Radvaz: This is like R. Akiva. Perush ha'Mishnayos explained like Chachamim! Perhaps the Halachah follows Chachamim, but nowadays that we are never Tahor, it is better to separate Tamei Chalah.

Or Some'ach: The Rambam rules like R. Akiva. The Ramban (Hilchos Chalah after Bikurim, 27b) rules like Chachamim against R. Akiva. It seems that this is because we say (Bechoros 3b) that Rav Mari bar Rachel's flock died out because he used to be transfer ownership of the ears to Nochrim, which totally uprooted the Kedushah. Chachamim (Bechoros 33b) permit letting blood from a Bechor sick due to excess blood, and compares this to causing Vadai Tum'ah to Terumah that is Safek Tamei. Just like it is better to make a Mum than to totally uproot the Kedushah, it is better to separate Tamei Chalah than to totally uproot the Mitzvah, like R. Akiva. Nowadays, one may blemish a Bechor before it is born (35a), to avoid stumbling in sin. This is better than selling to a Nochri, which totally uproots the Kedushah. The Rashba says that nowadays that there is no Taharah, it is better to take Tamei Chalah, lest the Mitzvah of Chalah be forgotten.

Tosfos (4a DH v'Chi): We say (Eruvin 32b) that it is better that a Chaver transgress a small matter so that an Am ha'Aretz not eat Tevel through the Chaver. When the Isur is not through Ploni, we do not tell Ploni to do even a small Isur to save another from a severe Isur. Further, we can say that one may not do a light sin to save someone else who transgressed from a bigger sin (but one may to save someone blameless). This explains why Kohanim may offer for a Mechusar Kipurim after the Tamid on Erev Pesach, and a Kohen may tear off a wart on another Kohen to enable him to serve.

Note: Seemingly, the Mechusar Kipurim was negligent for not offering his Korban before the Tamid!

Tosfos (48a DH Chet): One may not transgress the Lav of burning Shirayim of a Chatas for the sake of an Olah, but it is permitted for the sake of a Chatas. We do not transgress breaking a bone of Pesach in order to merit eating the marrow inside (Pesachim 85a), for eating does not fix the Korban as much as burning the limbs (of an Olah). Even fixing a Korban is permitted only if it was already slaughtered. If Shtei ha'Lechem was lost, one may not slaughter the lambs Lo Lishmah. If a Todah was mixed with a Temurah, one may not transgress even Chulin b'Azarah in order to fix it (Menachos 80a). However, if Dam ha'Nefesh was mixed with Pasul blood, we do not put it on the Mizbe'ach to merit with the Kosher blood (Zevachim 78a)! That is a decree, like it says there. If a Kometz was mixed with the Shirayim, we do not sin (burn) the Shirayim to fix the Kometz (Menachos 23a), because they have different names. (It is like sinning for the sake of a different matter.)

Ritva (Eruvin 32b): In Shabbos (4a), we do not tell someone to sin to help another. Even Rebbi agrees. There, Beis Din does not tell him, but he may do so by himself. If a Kohen got a wart that disqualifies him, another Kohen may bite it off on Shabbos (Eruvin 103b). We can say that enabling Kohanim to do Avodah is different. Some say that we do not tell someone to sin to help another who sinned. Even if he put the bread in b'Shogeg, he should have thought (more, and refrained). It is better to say that we do not tell someone to sin to help another avoid a Korban.

Tosfos (Yoma 29b DH Ela): If a wart was found on one Korban Pesach, all the groups are exempt from Pesach Sheni. The Gemara suggested bringing a Korban and stipulating that if they were already Yotzei, it is a Shelamim. Pesach must be after the afternoon Tamid. Shelamim is Pasul then! Do not say that Pesach, which has Kares, overrides Hashlamah. Here there is no Kares, since for each group, the majority says that they were Yotzei. Even if Shelamim after the afternoon Tamid were Kosher, the Torah forbids offering it then. If Pesach Sheni can be brought before the afternoon Tamid, the suggestion is understood.

Poskim

Shulchan Aruch (OC 254:6): If one stuck a dough in an oven on Shabbos, even if he was Mezid, he may remove it before it bakes, to avoid a Chiyuv Skilah.

Bach (6): Only he may remove it and transgress an Isur mid'Rabanan, to avoid a severe Isur. Others may not. We do not tell Ploni to do a light Isur to save Almoni from a severe Isur, since the Isur was not through Ploni, and Almoni was negligent.

Magen Avraham (21): Really, it primarily depends on whether Almoni was negligent.

Mishnah Berurah (40): Even if Almoni was Shogeg, others may not remove it, even if he is not here to tell him. Eliyahu Rabah says that even in our ovens, another may not remove it, because the dough is Muktzeh. This refers to baking. If Almoni forgot and left a pot near the fire, others must remove it to save Almoni from Isur. There is no Isur to remove a pot, therefore, it is a Mitzvah to save from sin.

Note: Surely this is when the food in the pot is ready to be eaten. If not, it is Muktzeh!

Kaf ha'Chayim (55): Tosfos Shabbos says that since we are concerned for Muktzeh, when he himself removes the dough, he should intend to feed it to dogs. However, the Gemara connotes unlike this.

Shulchan Aruch (YD 324:14): One may not make a dough less than a Shi'ur in order to uproot the obligation to take Chalah.

Beis Yosef (DH Asur, citing the Rashba): Even though we hold that one who cannot make a Tahor dough should make small doughs, nowadays one should not, lest the Mitzvah be forgotten.

Taz (17 and Shach 25): If he intends for a different reason, it is permitted, e.g. he has only a little dough. On Pesach, one should make small doughs (lest they become Chametz), and join them to become obligated.

Shach (25): The custom is that on Erev Shabbos, women try to bake a Shi'ur obligated in Chalah.

Meshiv Davar (2:44): Chachamim hold that one makes a small dough, for we do not tell someone to sin in order to gain. The Rambam rules like R. Akiva, for in Menachos we hold that one should sin in order to gain in the same matter. What is the source that the Halachah does not follow R. Yochanan, who disagrees? Tosfos questioned the suggestion to bring Pesach Sheni on condition. It seems that the Gemara suggested to slaughter it before the Tamid, like a Shelamim. One may sin with Pesach in order to fix Pesach. The Gemara asked that it will not be a proper Shelamim, for one may not sin with Shelamim in order to fix Pesach. We say in Menachos that one may transgress with Chatas for the sake of Chatas, but not for the sake of an Olah. This is why the Rambam rules like R. Akiva against Chachamim.

Note: We sin with birds (offer them after the Tamid for a Mechusar Kipurim) for the sake of Pesach!