Gemara

(Mishnah): A (healthy) baby is never circumcised after day 12.

Shabbos 137a (Mishnah): If he was born Bein ha'Shemashos (Safek day, Safek night), we must wait until day nine to circumcise. (We call the afternoon before the birth day one.)

If day nine is Shabbos, we must wait until day 10;

If the day after Shabbos is Yom Tov, we must wait until day 11;

We must wait until day 12 if the two days of Rosh Hashanah follow Shabbos.

130b (Beraisa): Just like one may not bring a Milah knife via Reshus ha'Rabim, it is forbidden via roofs, Chatzeros and Karfifos (open areas).

133a (Mishnah - R. Akiva): The general rule is, needs of Milah override Shabbos only if they cannot be done before Shabbos.

(Mishnah): If one did not grind the cumin before Shabbos, he chews it with his teeth and puts it on. If he did not mix wine and oil (this was used to heal) before Shabbos, each is put by itself;

134a (Beraisa): Things that may not be done for Milah on Shabbos may be done on Yom Tov, i.e. grinding cumin and mixing wine and oil.

Eruvin 68a: A case occurred in which the hot water prepared for a Milah spilled. Rava told people to ask the mother if she would like hot water (to drink). If she does, they may ask a Nochri to heat for her and the baby.

Rishonim

Rambam (Hilchos Milah 1:15): Any case that is not Docheh Shabbos is not Docheh Yom Tov Rishon, but it is Docheh Yom Tov Sheni, but not either day of Rosh Hashanah. Similarly, Milah not in its time is not Docheh either day of Rosh Hashanah.

Rambam (2:10): Machshirei Milah are not Docheh Yom Tov, even if it is the eighth day, since they can be done from before Yom Tov. We learn from a Kal va'Chomer. Machshirei Milah are not Docheh Shevus (mid'Rabanan stringencies of Shabbos). Surely they are not Docheh a Lav in the Torah!

Ramban (Shabbos 130b DH Ha): Someone permitted Yisre'elim to do needs for a Milah on Yom Tov Sheni. He learned from a Kal va'Chomer from burial. Chachamim made Yom Tov Sheni like a weekday regarding burial. He erred. We are more lenient to permit burial than Milah. Chachamim permitted Karmelis on Shabbos for the sake of burial, but not for the sake of Milah. They forbade burying through Nochrim, due to honor of the Mes. This is surely true according to the Rif, who forbids bringing a Milah knife through a Nochri via Reshus ha'Rabim. (The Ge'onim permit.) The Tosafists concluded like the Rif, based on Eruvin 68a. If the mother did not want hot water, they could not ask a Nochri to heat for the Milah. This is an absolute proof. Even if one will say that the text should say 'if she wants hot water, they may heat for her and the baby' (i.e. it does not say 'through a Nochri'), this must mean through a Nochri. Firstly, a Yisrael may not do Melachah for a Yoledes (one who gave birth) within 30 days (but after seven days). Also, when a Yisrael may do Melachah (for Piku'ach Nefesh), he may not add for a child (for Milah), like we say in Menachos (64a). If doctors say that a Choleh needs two figs (one may not cut a branch with three, if there is a branch with two).

Note: Tosfos (Gitin 8b DH Af) says that it is difficult to say that Yisraelim may do Melachah for a Yoledes within seven 24-days, and it was still within this time from the birth.

Ramban: Some disagree, for if one buys a city in Eretz Yisrael, we may write the document through a Nochri, for the sake of settlement of Eretz Yisrael. This shows that one may tell a Nochri to do a Melachah for the sake of a Mitzvah. I answer that there, all of Yisrael benefit, that Eretz Yisrael will not be desolate. Also, we cannot weigh Mitzvos. We do not know their reward, so we cannot say that if we permit for one, we must permit for another particular Mitzvah. We find that Yisre'elim may do Shevusim for the sake of a Mes, but not for Milah, even though the Aseh of Milah is more explicit in the Torah (than the Aseh of burial), and 13 covenants were made over it. In some places, Chachamim did not forbid a Shevus if it would cause a loss, e.g. a gutter pipe. (If it became clogged on Shabbos and the water is dripping into the house, one may covertly fix it with his foot - Kesuvos 60a. This is a Melachah in an unskilled way.) We do not permit every Shevus due to loss. We cannot compare one case to another.

Ramban (DH v'Chen): I say that we permit telling a Nochri to do even a Shevus only for Milah, since it can override Shabbos, but not for other Mitzvos.

Rosh (Teshuvah 26:5): Milah not in its time does not override Yom Tov Sheni of Galus (Chutz la'Aretz). The Mishnah says that if a baby was born Bein ha'Shemashos, so the Milah is postponed, it does not override either day of Rosh Hashanah. The same applies to Yom Tov Sheni of Galus. The Tana was in Eretz Yisrael, therefore he discussed Rosh Hashanah.

Poskim

Shulchan Aruch (YD 266:8): If a boy was born Bein ha'Shemashos, we count eight days from the night and circumcise him on day nine. We do not transgress even Yom Tov Sheni of Chutz la'Aretz due to the Safek.

Shach (8): The Rambam's opinion is primary, that it overrides Yom Tov Sheni of Galus. Also Semag and Kolbo say so. The Bartenura (Shabbos 19:5) explains that both days of Rosh Hashanah are one Kedushah (i.e. but Yom Tov Sheni of Galus is different). The Rambam and his followers say that other matters that do not override Shabbos, override Yom Tov Sheni of Galus.

R. Akiva Eiger: Why doesn't it override due to a Sefek-Sefeka (two doubts)? Perhaps it is the proper time, and perhaps today is not Yom Tov! The Mechaber (OC 497:4) perhaps a Safek about Muchan (whether stranger was prepared before Yom Tov) due to such a Sefek-Sefeka. Perhaps there it is a Sefek-Sefeka about a mid'Rabanan law, and here it is mid'Oraisa. However, those who forbid there forbid because it is Yesh Lo Matirim (it is totally permitted after Yom Tov). This does not apply to a Mitzvah, for it is proper to do the Mitzvah promptly. Besamim Rosh (88, 253) permits Milah not in its time on Yom Tov Sheni. In any case we can rely on this when it is a Safek if Yom Tov Sheni is the eighth day. We do not find that anyone argues about this. Later, I saw Be'er Heitev; the Av Beis Din Berlin was astounded at the Shulchan Aruch, and permits on Yom Tov Sheni if it is a Safek whether it is day eight. Tashbatz (3:284) forbids. He brought a proof from the Mishnah, and says that the same applies to Yom Tov Sheni of Galus. I do not understand. The Mishnah discusses one who was born Bein ha'Shemashos of Erev Shabbos, and Yom Tov (Sheni) is on Monday. It is surely not in its time. There is no source to forbid if it is a Safek whether Yom Tov Sheni is day eight.

Noda bi'Yehudah (1 OC 30): I disagree witrh the Mechaber. Usually he seeks reasons to rule like the Rambam even when many disagree. In any case he does not totally omit the Rambam's opinion, and brings 'some say that...' Here, the Rambam permits Yom Tov Sheni of Galus even if it is Vadai after day eight. The Beis Yosef brought only one opposing opinion, of the Rosh. The Shach rules like the Rambam. The Shulchan Aruch forbids even when it is a Safek Milah bi'Zmano (in its time). The Rosh never forbade this! He discusses a baby born on Erev Shabbos. Yom Tov is after the following Shabbos. It is surely after day eight. If it is a Safek bi'Zmano, we permit due to a Sefek-Sefeka. Perhaps the Mechaber forbids the Sefek-Sefeka due to Davar she'Yesh Lo Matirim, like the Rema says (YD 110:8). However, the Mechaber (497:4) permits Safek Muchan due to Sefek-Sefeka (even though Yesh Lo Matirim). The Rema did not comment. The Shach answered that Yom Tov Sheni of Galus is more lenient, for it is only a custom. Alternatively, we are more stringent about a mixture, for surely there was Isur. In either case, we should permit Milah! I say that the Rema holds like the Ran, who forbids Sefek-Sefeka when Yesh Lo Matirim for an Isur Torah, and permits for an Isur mid'Rabanan. If so, also the Mechaber does not contradict himself.

Hagahos Betzalel Luria: The Re'em said that a Tumtum (Safek male, Safek female) cannot be Motzi another Tumtum, because the Sefekos are in two bodies. Here also, one Safek is in the child, and one is in the day.

Hagahos Halachah l'Moshe: Surely delay of a Mitzvah is a loss. We are not stringent about Davar she'Yesh Lo Matirim even regarding a Safek loss, for we cannot say 'rather than eat it amidst (Safek) Isur, eat it amidst (total) Heter.

Avnei Nezer (OC 392): May one leave the Techum for Milah on Yom Tov Sheni? One may not leave the Techum for Shofar (Rosh Hashanah 32b), because an Aseh is not Docheh a Lav and an Aseh. We hold that it is Docheh a Lav and an Aseh that do not apply to everyone! Yom Tov Sheni applies only to Bnei Chutz la'Aretz. The Aseh of Milah applies to everyone. It does not apply to women only because they have no Orlah. Slaves have only the Mitzvos of women, and Milah applies to them! (This is why an Androginus (who has male and female genitals) is like a definite male for Milah, since he has Orlah.) However, perhaps this is wrong, for even Bnei Eretz Yisrael are commanded about Yom Tov Sheni if they would go to Chutz la'Aretz.

Note: Tosfos (Kidushin 34a DH Ma'akah) says that the Aseh of Yom Tov does not apply to everyone, for he exempts women.

Avnei Nezer: Also, one does not fulfill the Mitzvah of Milah at the time he transgresses Techumim. The Ritva and Ran say that the Gemara could have said that this is why it is not Docheh. Even according to the Lechem Mishneh (Hilchos Shofar 2:6) who requires both of these together, this is only for other Isurim, but not for Techumim, like I explained in the introduction to Iglei Tal. Actually, we can permit. The Rambam permits Milah not in its time on Yom Tov Sheni of Chutz la'Aretz. The Beraisa forbids only Yom Tov Sheni of Rosh Hashanah, which connotes that Yom Tov Sheni of Chutz la'Aretz is permitted. The Rosh disagrees.

Avnei Nezer (5,6): The Noda bi'Yehudah rules like the Rosh. Why should we be more lenient about Yom Tov Sheni of Galus regarding something that does not depend on one long day (48 hours of Kedushah)?! Regarding a Mes, we are as lenient about Rosh Hashanah as we are about Yom Tov Sheni of Galus. This is astounding. We are lenient about painting the eye on Yom Tov Sheni of Galus, but not on Rosh Hashanah! There is no proof from a Mes. It is Docheh even Yom Tov Rishon of Rosh Hashanah!