Gemara

R. Yitzchak bar Bisna lost the keys to the Beis Medrash in a Reshus ha'Rabim on Shabbos. R. Pedas counseled to get minors to play there, for if they find the keys they will bring them.

He holds that if a minor is eating forbidden food (or doing any other Isur), Beis Din need not stop him.

Support (Beraisa): A man should not tell a minor to bring to him a key or a stamp, but he may allow the minor to uproot or throw things.

Rejection (Abaye): The minor may uproot from a flowerpot without holes (what grows inside is not considered attached to the ground) or throw in a Karmelis (these are forbidden only mid'Rabanan).

Question (Mishnah): If a minor comes to extinguish a fire on Shabbos, we tell him not to extinguish, since we are commanded that he rest.

Answer (R. Yochanan): It is forbidden when the minor does so to please his father.

Support (Beraisa): If the son of a Chaver (one trustworthy about Ma'aser) often goes the house of his mother's father (an Am ha'Aretz), we are not concerned lest they feed him Tevel. If the father finds food in his son's hand, he is not responsible.

Rejection (R. Yochanan): This is no support. We are more lenient about Demai (doubtfully tithed produce).

Question: This implies that if it was definitely Tevel, he would need to Ma'aser it. Above, R. Yochanan answered 'he does so to please his father' (but we need not stop minors who transgress for their own benefit)!

Answer: R. Yochanan is in doubt what is the Halachah. He rejects the proofs for both opinions.

Support (Beraisa): A child may nurse from a Nochris or a Tamei animal. We are not concerned that he suckles something detestable and forbidden. One may not feed to a child Neveilah, Treifah, Tamei rodents or vermin;

The child may nurse from them even on Shabbos. An adult may not;

Aba Sha'ul says, we used to nurse from Tahor animals on Yom Tov.

Conclusion: The Beraisa says that we are not concerned that he suckles Isur!

Rejection: It is permitted because the child is in danger if he does not nurse.

(Rav Huna brei d'Rav Yehoshua): We assume that a child is in danger if he lacks milk.

Question: Aba Sha'ul said that they used to nurse from Tahor animals on Yom Tov. What is the case?

If there is a danger, it should be permitted even on Shabbos. If there is no danger, it should be forbidden even on Yom Tov!

Answer: The case is, they were in pain. He holds that nursing is Mefarek (extracting) in an unskillful manner, which is forbidden only mid'Rabanan.

On Shabbos one (who is Mefarek normally) is liable to stoning, therefore Chachamim decreed (against unskillful Mefarek, e.g. nursing, even though he is in pain). Skilled Melachah on Yom Tov is only a Lav, so they did not decree to forbid one in pain.

Rishonim

Rif and Rosh (Shabbos 45b and 16:10, citing the Mishnah): If a minor comes to extinguish a fire on Shabbos, we do not allow him, since we are commanded that he rest.

Ran (DH Masnisin): We hold that extinguishing is only mid'Rabanan, for the Halachah follows R. Shimon who exempts Melachah she'Einah Tzerichah l'Gufah. The Rashba holds that we may feed to a minor (or prompt him to do) an Isur mid'Rabanan! However, this is only for his benefit, not for ours.

Rambam (Hilchos Ma'achalos Asuros 17:27): If a minor did Melachah on Shabbos, Beis Din is not commanded to separate him, for he does not have Da'as.

Kesef Mishneh: Even though R. Yochanan was unsure, since R. Pedas was sure and acted upon this we follow him, in spite of R. Yochanan's doubt.

Rambam (Hilchos Shabbos 12:7): If a minor comes to extinguish a fire on Shabbos, we do not allow him. This is when he does so to please his father. If he does so for himself, Beis Din is not commanded to stop him.

Rambam (24:11): If a minor did something forbidden on Shabbos mid'Rabanan, e.g. he uprooted from a flowerpot without a hole or carried in a Karmelis, Beis Din need not stop him. If his father lets him do so, we do not protest.

Question (Magid Mishneh): This implies that Beis Din must stop a minor who is doing a Melachah mid'Oraisa. This is not true! The Gemara (Yevamos 114a) concludes that Beis Din does not stop him, and the Rambam himself says so above (12:7) and in Hilchos Ma'achalos Asuros 17:27)!

Answer (Kesef Mishneh): Even if it was an Isur mid'Oraisa Beis Din would not stop him, but then Beis Din would protest against his father if his father did not rebuke him.

Rambam (Hilchos Ma'achalos Asuros 17:28): Even though Beis Din is not commanded to stop a minor, his father must rebuke him and separate him in order to train him in Kedushah - "Chanoch la'Na'ar Al Pi Darko".

Magid Mishneh: If he father does not rebuke him, Beis Din is commanded to invigorate the father to rebuke his son to stop him.

Rosh (Shabbos 1:1): Beis Din is commanded to stop a minor from eating Isur, all the more so we may not help an adult transgress!

Nimukei Yosef (ibid.): The Halachah is that Beis Din is not commanded to stop a minor. The Gemara here and in Gitin (55a) asks as if this is known, and R. Pedas acted upon it in practice. No Amora disagrees. We answered every question against him. The case of extinguishing to please his father is when he said father is there and indicates that he desires this. This is like telling him to extinguish; we do not feed Isur to a minor (or overtly cause him to transgress). The Rashba holds that for mid'Oraisa Isurim we need not stop minors, but we do not feed them. We are more lenient about mid'Rabanan Isurim and may feed them to minors. The Rambam forbids feeding even Isurim mid'Rabanan.

Poskim

Shulchan Aruch (OC 334:25): If a minor comes to extinguish a fire on Shabbos, one must protest.

Beis Yosef (DH v'Amai and 343 DH v'Efshar): The Mishnah says that we do not let a minor extinguish. The Gemara tried to prove from that Beis Din is commanded to stop a minor; R. Yochanan rejected this by saying that the minor does so to please his father. Rashi explains that his father is there and the minor understands that his father desires this. It is as if the father commands him. When keys were lost R. Pedas suggested bringing minors who did not know that he wanted the keys. In such a case we may let a minor extinguish. However, it seems that one must always protest, for a child knows that his father wants him to extinguish. If the child is doing an Isur unaware that this will prevent damage, we may let him, even if he knows that his father wants to avoid damage. R. Yerucham connotes that if the child acts to please any adult we must stop him.

Beis Yosef (DH v'Efshar): The Rif and Rosh brought the Mishnah without R. Yochanan's explanation. Perhaps R. Yochanan was unsure about the Halachah, so also we are unsure. A support for this is that the Rosh (1:1) says that Beis Din is commanded to stop a minor. However, perhaps the Rif and Rosh brought only the Mishnah because R. Yochanan's answer was that we always assume that he extinguishes to please his father.

Shulchan Aruch (343:1): If a minor is eating Neveilos, Beis Din is not commanded to stop him, but his father must rebuke him and separate him (Rema - from an Isur mid'Oraisa). One may not accustom him to Mechalel Shabbos or Mo'ed, even in Isurim mid'Rabanan.

Magen Avraham (1): Beis Din is commanded to stop a minor if it will cause ruin, e.g. if adults hear him blow a Shofar on Shabbos and they will do similarly.

Rema: Some say that this refers to a minor below the age of Chinuch, but once he reached Chinuch we must stop him. Some say that Chinuch applies only to the father, not to Beis Din.