Gemara

63b (Rav Chisda): If a man says that his son (or daughter) is over nine (or three) years old, (so the minor's Bi'ah is considered Bi'ah), he is believed for Korbanos (e.g. if an Ervah had Bi'ah with the minor b'Shogeg), but not for lashes or death.

Support (Beraisa): If a man said 'my son (or daughter) is Bar (or Bas) Mitzvah', he is believed for (the minor's) vows, Cherem, Hekdesh, and Erchin (they take effect). He is not believed for lashes or death if the child transgressed.

Yevamos 33a: If a Zar brought two hairs (after 13 years, and became an adult) on Shabbos, the Isurim of Avodas Zar and Melachah on Shabbos take effect on him at the same time, when he becomes an adult.

Nidah 46a (Rav Huna): If a Mufla Samuch l'Ish (a minor in the year before adulthood) was Makdish something and ate it, he is lashed;

It says "Ish Ki Yafli Lindor Neder," and it says "(Ish Ki Yidor...) Lo Yachel Devaro" -- whoever is an "Ish" for the former (his vows take effect) is an Ish for the latter (he is liable for transgressing them), and vice-versa.

Support (Rav Huna bar Yehudah - Beraisa) Suggestion: A minor is like an adult for... Lo Yachel (the Lav to transgress oaths and vows). Perhaps a Korban Me'ilah is brought for benefit from Hekdesh of a minor!

Rejection: "Zeh ha'Davar" (for Me'ilah, he is not like an adult. However, he is lashed for Lo Yachel!)

Rejection: He is commanded about Lo Yachel like an adult, but he is not lashed.

Question: In any case this is difficult! If Mufla Samuch l'Ish is mid'Oraisa, he should be lashed. If it is mid'Rabanan, he should not be forbidden!

Answer: Mid'Rabanan, the minor's overseers must stop him from transgressing.

Inference: Beis Din (i.e. all adults) must stop minors from transgressing!

Rejection: No. The case is, the minor was Makdish food and adults ate it.

Question: This is Like R. Yochanan and Reish Lakish, who say that an adult is lashed for (benefit from) a minor's Hekdesh. According to Rav Kahana, who says that an adult is not lashed for a minor's Hekdesh, how can we answer?

Answer: He is lashed mid'Rabanan. What we expounded from "Ish Ki Yafli" is only an Asmachta.

Yevamos 113b: 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 Beis Din need not stop a minor from eating (or doing) an Isur.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 17:27,28): If a minor ate a forbidden food or did Melachah on Shabbos, Beis Din need not stop him, for he lacks Da'as. Even so, his father must chastise him and stop him, to train him in Kedushah and separation - "Chanoch la'Na'ar Al Pi Darko."

Lechem Mishneh: This is the Rambam's reasoning. It is proper.

Nimukei Yosef (Yevamos 42a DH Garsinan): The Rashba says that since we train minors to do Mitzvos, surely we must prevent them from Isurim! However, the Gemara suggested that Beis Din is commanded about a Mufla Samuch l'Ish, and rejected this Perhaps Chinuch applies only to Mitzvos Aseh, but not to Lavin! The Ritva forbids feeding an Isur mid'Rabanan even to a minor below Chinuch. The Halachah is that Beis Din need not stop a minor. The Gemara in Yevamos and Gitin (55a) assumes so, and R. Pedas acted upon it in practice. No Amora disagrees. We answered every question against him. The Rashba holds that for mid'Oraisa Isurim we need not stop minors, but we do not feed them. We may feed mid'Rabanan Isurim to minors. The Rambam forbids feeding them even Isurim mid'Rabanan.

Radvaz: If the father slackens from doing so, Beis Din must invigorate him to chastise his son to stop him.

Mishneh l'Melech: We answered that the minor's overseers are commanded. Rashi said that this is like the opinion that Mufla Samuch l'Ish is mid'Rabanan, and like the opinion that Beis Din must stop minors from transgressing. No one says that Beis Din must stop them from an Isur mid'Rabanan! Perhaps Rashi made a mere comparison to that argument (Rashba). Rashi explained like this, for if Mufla Samuch l'Ish is mid'Oraisa, the minor himself is lashed. However, we hold that Mufla Samuch l'Ish is mid'Oraisa, and he is not lashed, for he is a minor regarding punishments! It seems that the conclusion is that even if Mufla Samuch l'Ish is mid'Oraisa, the minor is not commanded.

Mishneh l'Melech: Why are only the overseers commanded? No one may feed Isur to a minor! Perhaps this is like the Nimukei Yosef, who says that the one who exempts Beis Din from stopping minors from transgressing permits others (not his overseers) to feed to him an Isur mid'Rabanan.

Tosfos (64a DH Ne'eman): Normally, one witness is believed only about something b'Yado (in his control). Since the minor will become an adult, Chachamim considered it to be b'Yado. They did so only for the father, for it is incumbent on him to inform people when his minor is an adult.

Kovetz Shi'urim (Bava Basra 413): Shiltei ha'Giborim (Kidushin 32a DH v'Chen) says that a man is believed to say that his minor is Bar or Bas Mitzvah due to "Yakir" (the Torah believes him to say who is the Bechor). He is believed for everything except for punishments. According to Tosfos, the father is not believed for Chalitzah, for it is Devar sheb'Ervah, but when he is believed due to Yakir, he is believed even for punishments, just like he is totally believed about his daughter's Kidushin. The Nimukei Yosef (Bava Basra 56a DH Tanan) says that he is believed against witnesses. If so, surely he is believed for punishments.

Hagahah: The Nimukei Yosef said only that if witnesses say that Reuven was born before Shimon, and Yakov says 'Shimon is my Bechor', we establish Reuven to be a Mamzer. We do not believe the father against witnesses!

Kovetz Shi'urim (ibid.): Normally, one witness is believed for punishment. If one says 'this is Chelev' and another eats it, lashes are a direct result of the Isur. We do not divide the Ne'emanos (for what he is believed). Regarding a minor, we divide it. The reason he is not punished is not because he is not commanded. Rather, even if he is commanded, he is not punished. Therefore, we can divide it. When one witness says that Leah's husband died, he is believed even for the Kesuvah, for this directly results from her Heter to remarry (whereas inheritance does not).

Terumas ha'Deshen (62): Even a minor who reached the age of Chinuch is not liable at all, even a moment before he brings two hairs. If a Zar became an adult on Shabbos, the Isurim of Avodas Zar and Melachah on Shabbos take effect on him at the same time (Yevamos 33a). Also,. Beis Din need not stop a minor from eating Neveilos. If a minor wounded or damaged he is exempt, even when he matures, for he is not liable to punishments (Rosh Bava Kama 8:9). We derive that Mitzvos and Chukim do not apply to anyone exempt from punishments.

Question (Kuntres Divrei Sofrim (in Kovetz Shi'urim 2, 21)): Some say that a minor (past the age of Chinuch) is himself obligated in Mitzvos, unlike Rashi (Berachos 48). According to the Ramban, there is no Mitzvah mid'Oraisa to obey Chachamim. If so, what obligates a minor?

Answer (ibid., 22): All must do Hash-m's will. We know that Chachamim's commands are His will. Minors are exempt minors only from Torah Mitzvos.