Gemara

(Beraisa): "Seh l'Veis Avos" teaches that a man brings and slaughters Pesach for his minor children with or without their consent.

Nedarim 36a (R. Zeira): The Drashah "Seh l'Veis Avos" is not mid'Oraisa. (It is an Asmachta mid'Rabanan.)

Rishonim

Rambam (Hilchos Temidim u'Musafim 7:25): One must bless every night on Sefiras ha'Omer before counting. If he counted without blessing, he was Yotzei, and he does not bless.

Hilchos ha'Gra u'Minhagav (74): Perhaps there is no Chinuch of a minor for Keri'as Shema and Sefiras ha'Omer, since he is normally sleeping after Tzeis ha'Kochavim. An Ashkenazi minor can bless, just like a woman blesses on Mitzvos that she herself is not commanded. It is not clear why a Sefardi minor may bless. However, even if it is a Berachah l'Vatalah, he can bless, so he will do so after he matures. This requires investigation. The Rambam supports this [that without blessing, it is not proper Chinuch - PF.]

Tosfos (88a DH Seh): If "Seh l'Veis Avos" is not mid'Oraisa, how may one feed Korban Pesach to his children? Even though if a child is eating Nevelos, Beis Din is not commanded to stop him, one may not overtly feed him! The Ri answers that we learn the Isur to feed from "Lo Sochlum." It is forbidden only in cases like Sheratzim and Nevelos, but not in a case like Korban Pesach, in which there is Chinuch for a Mitzvah.

Tosfos (Rosh Hashanah 33a DH Tanya): Rashi explains that if a child reached the age of Chinuch, we engage him with a Shofar until he learns [how to blow it], even on Shabbos. Below the age of Chinuch, we do not stop him. This is astounding. Why is it permitted above the age [of Chinuch]? There is no Mitzvah on Shabbos! When he is below the age, why don't we give to him l'Chatchilah? Even above the age, one may engage him in it, so he will learn! There However, we could explain that one need not teach him, since he did not reach the age, but there is no Isur to do so. I explain oppositely. If a child reached the age, we do not stop him, but l'Chatchilah we do not tell him to blow, since there is an Isur mid'Rabanan. One need not stop him, since if it were not Shabbos, this would be the Mitzvah of the day. We engage a child below the age, i.e. one may [for he has no Isur]. Some texts say "until he learns, even on Shabbos. We do not stop him on Yom Tov. On Shabbos, we stop him. In Yevamos (114a), a Beraisa forbids telling a child to bring a key. Abaye established this to discuss an Isur mid'Rabanan. This shows that one may not overtly cause a minor to transgress even mid'Rabanan! Even so, one may engage him with a Shofar even on Shabbos, because sometimes there is a Mitzvah to blow. It is unlike other Isurim. We say in Erchin (2b) that "everyone is obligated to blow the Shofar" includes a minor who reached the age of Chinuch, like the Mishnah says that we do not stop him from blowing.

Poskim

Shulchan Aruch (OC 489:8): If one forgot to count a day, he counts the remaining days without a Berachah.

Chasam Sofer (1:172 DH ul'Inyan): If a child became Bar Mitzvah on Motza'ei Yom Kipur, he was exempt from fasting. Likewise, he is exempt from adding to the fast at the end of the day even if he brought two hairs. The Tosefes is not part of the fast itself, to say that it extends into the night. There is no Kares or lashes for one who eats then. It is a mere Isur that Chazal expounded to add from Chol to Kodesh. If one was not obligated to treat the day like Kodesh, he is exempt from the Tosefes. One cannot make an addition without principal! This is unlike a Choleh who ate throughout Yom Kipur, and became healthy at the end of the day. Tosefes applies to him. The day was Kodesh for him, just he was permitted to eat due to Piku'ach Nefesh. He conducted with Kedushah, like Hash-m commanded, and now he adds to it. The minor did not treat it like Kodesh mid'Oraisa, so he has no Tosefes mid'Oraisa.

Chasam Sofer: However, if a minor became Bar Mitzvah in Motza'ei Shevi'is, he must treat with Kedushah attached Peros that grew during Shevi'is. There is very different. The Kedushah was on the land and the Peros. They received Kedushas Shevi'is, and from their Chol (Sheminis) we add to the primary Kedushah. They are forbidden also to the minor after he matures. Yom Kipur is different. Its time passed. There is a Mitzvah on people to add to the Kodesh. The minor had no Kodesh, so he has nothing to add.

Rebuttal (Mishpetei Uzi'el OC 22): Once he matures, he is commanded to observe all the Mitzvos, even if they are continuations of Mitzvos that began earlier.

R. Akiva Eiger (OC 186:1): If a minor ate to satiation during the day, and became Bar Mitzvah at night before the food was digested, perhaps he must bless Birkas ha'Mazon mid'Oraisa. Or, perhaps he is exempt, for when he ate, the Torah did not obligate him to bless. If he is liable, even if he blessed during the day, this does not fulfill his Torah obligation later. Chachmas Adam (153:1) asked about an Onen who ate and buried his Mes before the food was digested. There he was always obligated, just Aninus overrode other Mitzvos.

Ohr l'Tziyon (1:36): If a minor counted before his Bar Mitzvah, all agree that this is called Temimos (seven full weeks). Even if you will say that a minor's Mitzvah does not help for after maturity, presumably, just like we are commanded to train children in Mitzvos, so he will be used to observing Mitzvos, likewise if one matured and has half a Mitzvah to fulfill, mid'Rabanan he must complete it. I.e. even if he has no Torah Mitzvah, he is commanded mid'Rabanan, due to Chinuch. R. Akiva Eiger supports this. Why didn't he ask about one who did not bless when he was a minor? We must say that in such a case, surely he must bless at least mid'Rabanan. Rashi (Devarim 11:18) cites a Medrash that obligates doing Mitzvos in Chutz la'Aretz after Galus, so they will not be new to us when we return to Eretz Yisrael. This is like Chinuch regarding adults!

Magen Avraham (267:1): The Mordechai (Megilah 798) says that the opinion that Tosefes Shabbos is mid'Rabanan agrees that one is Yotzei with Kidush that he said before dark, since the Torah Chiyuv will come later. Why is this different than a minor, who cannot exempt an adult obligated mid'Oraisa in Birkas ha'Mazon, even though later the child will be obligated mid'Oraisa?

Note: Likutei Chaver ben Chayim answers that the child will not be liable for a long time, so perhaps he will die first, so he is not considered one who will become liable. I would add that his Chiyuv will not come by itself (he must eat). However, perhaps the Magen Avraham asks from a minor who will be Bar Mitzvah very soon, before the food is digested (see R. Akiva Eiger above). He is less apt to die beforehand than for an adult during Tosefes Shabbos! Perhaps a minor cannot be Motzi others because Arvus (mutual responsibility for all Yisre'elim) does not yet apply to him, even though soon it will!

Minchas Chinuch (306:6): The Magen Avraham holds that a Chiyuv mid'Rabanan can be Motzi a Torah Chiyuv. If so, what a minor counted before maturity, when he was obligated due to Chinuch, helps for after maturity.

R. Akiva Eiger (267:2): (If one slaughtered for a minor on Pesach Rishon, and he became Bar Mitzvah before Pesach Sheni, he is exempt from Pesach Sheni.) The Kesef Mishnah (Hilchos Korban Pesach 5:7, citing Ri Korkus, learns from a verse that allows including a minor on Korban Pesach. This) is unlike the Mordechai. (According to the Mordechai, in any case the minor was Yotzei, for he will be obligated later!)

Note: Perhaps Pesach Sheni is different than Kidush, for it is a new Mitzvah.

Or Some'ach (Korban Pesach 5:7 b'Sof): No one requires a minor who becomes Bar Mitzvah during the festival to delay bringing Olas Re'iyah and Chagigah until he matures. Rather, he was Yotzei with what he brought with Klal Yisrael when he was a minor, because he was obligated due to Chinuch.