Gemara

(Mishnah): Beis ha'Teme'os (where women stay when they are Nidah) of Kusim have Tum'as Ohel, for they bury Nefalim there.

57a - Question: What do Kusim expound (to justify not burying Nefalim)?

Answer: "Lo Sasig Gevul Re'acha Asher Gavlu Rishonim b'Nachalascha" -- only one with Nachalah (inheritance) has Gevul (burial).

Bava Basra 101a (Mishnah - R. Shimon): One who accepted to build a burial cave... makes 13 hollows for coffins... and two by the entrance, one on each side.

Question: How are the two Kuchin near the entrance positioned?

Answer: They are for Nefalim. (They are in the free space on the long side. They are very small.)

Sanhedrin 46b - Question: Is the reason for burial to avoid disgrace (of a rotting corpse), or for atonement?

The answer teaches about one who asked (before he died) not to be buried: if we are concerned for disgrace, we do not comply. If it is for atonement, since he does not want it, we honor his request.

47b (Beraisa): One may benefit from a new grave. If a Nefel was put inside, one may not benefit from it.

Kesuvos 20b (R. Chanina): Mounds that are close to the city or the road are Tamei, because women bury Nefalim in them...

Avodah Zarah 42a (Beraisa - R. Yehudah): A case occurred in which a slave dropped a Nefel (miscarriage) into a pit. A Kohen stood over it. Chachamim ruled that he is Tahor, for perhaps weasels dragged it away.

Inference: Even though there definitely was Tum'ah, and we are unsure if it was dragged away, we are lenient!

Rejection: She dropped a fetal sac in the pit.

Temurah 34a: Ashes of Nisrafim (things that must be burned) are permitted), ashes of Nikbarim (things that are buried) are forbidden.

Question (Beraisa): If flesh of a Mes crumbled, it is Tahor.

Suggestion: It is Tahor and permitted (b'Hana'ah).

Rishonim

Rambam (Hilchos Tum'as Mes 8:3): Mounds close to the city or the road are Tamei, because women bury Nefalim in them.

Rambam (9:2): If there is a pit in which Mezimim are cast, if one removes all the bones it is Tahor.

Rambam (Hilchos Evel 14:18): If one builds a grave for a Mes, it is permitted until a Mes is put in. Even if a Nefel is put in, it is forbidden.

Rosh (Mo'ed Katan 3:94): Evel Rabsi (1) teaches that we do not engage with a Nefel at all.

Poskim

Shulchan Aruch (YD 344:8): We do not eulogize or make a funeral for a Nefel.

Shach (6, citing the Tur): We do engage in needs of burial.

Rema (OC 526:10): One may not bury a Nefel on Yom Tov. We leave it until the next day.

Beis Yosef (DH v'Chen, citing Hagahos Maimoniyos Hilchos Milah 1:10): There is no Mitzvah to bury a Nefel. We find that they used to cast them to pits.

Bedek ha'Bayis: This refers to a Vadai Nefel. If it is a Safek, we bury it on Yom Tov.

Magen Avraham (20): What is Hagahos Maimoniyos' proof? Casting into a pit is burial! Also, in Avodah Zarah we say that the 'Nefel' was really a fetal sac. In Nidah, we explain why Kusim do not bury Nefalim. This implies that we bury them! Toras Kohanim expounds that a Kohen does not become Tamei for a Nefel. This shows that it is a Mitzvah to bury it, for a Kohen becomes Tamei only for needs of burial (YD 373). Also, we find that they used to make burial caves with hollows for Nefalim. Nidah 57 connotes that Bal Talin (the Isur to leave a Mes overnight) applies to Nefalim.

Machatzis ha'Shekel: Tosfos (56b DH Kovrin) explains that the Kusim intend to bury the Nefel afterwards. We must say that they bury it in the house beforehand due to Bal Talin.

Chavos Ya'ir (65): A Tosefta in Ohalos (Perek 16, based on the Mishnah 16:3) says that people used to cast Nefalim to pits. Women and Amei ha'Aretz consider this to be an abomination worthy of punishment.

Noda b'Yehudah (1 OC 16): The Beis Yosef says that we do not circumcize a Nefel even on Yom Tov Sheni, and we do not bury it. Since we bury it only after Yom Tov, obviously we do not circumcize it! The Beis Yosef needed to teach about a Safek Nefel. We may bury it on Yom Tov Sheni, for it is a Safek mid'Rabanan, and there is concern for Bal Talin. However, we cannot circumcize a Safek on Yom Tov. Even if the baby was viable, some say that Milah after death for the sake of dead Resha'im (lest the baby's foreskin be put on them). It is like a Sefek Sefekah, which overrides a majority (most babies are viable), so we are stringent even about Yom Tov Sheni. The Darchei Moshe holds that since we may not circumcize it today, a Sefek Sefekah permits leaving it until tomorrow. (Even if it is viable, perhaps the Milah is for its honor; one may leave a Mes overnight for its honor.) The Magen Avraham's proofs that there is a Mitzvah to bury a Nefel do not override Hagahos Maimoniyos. The Magen Avraham himself wavered and concluded 'this requires investigation'. I see no source from Nidah that Bal Talin applies to a Nefel. Logically, Bal Talin is due to the honor of the Mes. This does not apply to a Nefel. Even if there is a Mitzvah to bury it, perhaps burial overrides Yom Tov Sheni only when there is concern for Bal Talin mid'Oraisa.

Bach (YD Sof Siman 353, citing Ohr Zaru'a): If a baby died within 30 days, one need not take it out in a bier, rather, in a box. One may take it out in a bier. If it did not have a full term pregnancy, it need not be buried. One may cast it into a pit.

Suggestion: Tosfos Yom Tov (Shabbos 10:5 DH v'Chen) says that there is a Mitzvah to bury k'Zayis of a Mes. This implies that there is a Mitzvah to bury a Nefel!

Rejection (Noda b'Yehudah 1 YD 90): Burial does not depend on a Shi'ur, rather, on honor of the Mes.

Suggestion: Temurah 34a forbids ashes of Nikbarim. It asked why a Beraisa permits flesh of a Mes that crumbled. If there were no Mitzvah to bury a Nefel, perhaps the Beraisa permits because it discusses a Nefel!

Rejection (Noda b'Yehudah): There is no Mitzvah to bury Nikbarim, just there is no Mitzvah to burn them. We bury them so people will not benefit from them. The ashes are forbidden because there is no Mitzvah to bury them! The same applies to a Nefel.

Tzitz Eliezer (10:25:8): There is no Chiyuv to bury a Nefel. Eretz Chayim (YD 357) brings several sources that the custom is not to bury a Nefel within nine months, until after the mother recovers, lest it be dangerous for her. He cites Kapei Aharon, who brings a case in which Gedolei ha'Dor said not to be concerned for the custom, and to bury it (immediately). Since there is no basis for the custom is, we do not follow it. Also Darchei Teshuvah (YD 116:43) says that we are not concerned for what some women say about this, for it has no source in Chazal, the Ari Zal or Yisrael or Nochri doctors. However, several Poskim say that there is a Mitzvah to bury a Nefel, and Bal Talin applies. Mishmeres Shalom (100:13) says that this is only if the Nefel was (at least) five or six months, and has all the limbs. Maharsham (4:146, b'Sof) says that if she miscarried within three months (especially of she did not feel it), all agree that we need not bury it. The Yerushalmi mentions a custom to bury a Shilya. It is like a security (that the land will eventually get the child), and an omen for long life. Sedei Chemed (Kuf, 37) says that Nechmad l'Mar'eh criticizes women who simply dispose of it. He does not obligate burying it, just it is better to bury it.

Gesher ha'Chayim (1:16:2:3): I did not see a custom to bury a Shilya. The Mitzvah of burial does not apply to a Nefel, therefore, one may not bury it on Yom Tov, but we bury it because one may not benefit from it, or due to concern for Tum'ah.