The Shulhan Aruch (Orah Haim 331:6) establishes the rule that although circumcision is performed for a baby on his eighth day even if it is Shabbat, "Machshireh Mila" – preparatory stages of circumcision – do not override the Shabbat prohibitions. This means that anything needed for the Berit which could have been done before Shabbat may not be done on Shabbat if it entails an act forbidden on Shabbat. The classic example is bringing the knife through a public domain. Since the knife could have been brought before Shabbat to the place where the Berit would be taking place, this is not permitted on Shabbat, even if as a result the Berit will have to be delayed until Sunday. This rule applies even to Rabbinic violations, such as carrying a knife through a Karmelit – a public domain through which carrying is forbidden only by force of Rabbinic enactment. The Rabbis applied their enactment even to a situation where carrying is necessary for a Berit Mila, and thus the knife may not be carried even through a Karmelit for the sake of performing a circumcision on Shabbat.
However, the Shulhan Aruch writes that it would be permissible on Shabbat to ask a non-Jew to bring the knife through a Karmelit for the sake of a Berit Mila that needs to be performed that day. Since asking a non-Jew to perform a Melacha on Shabbat is forbidden only Mi’de’rabbanan (by force of Rabbinic enactment), and carrying through a Karmelit – even for a Jew – is forbidden only Mi’de’rabbanan, this may be allowed for the sake of the great Misva of circumcising a baby on his eighth day.
The more common question that arises is whether the baby may be carried to a synagogue on Shabbat for his Berit. Very often, the young couple live in a small apartment, where they would be unable to host a large assembly of people for the Berit, and they wish to have the Berit performed in a synagogue or in some other large facility. A large audience for a Berit fulfills the dictum of "Be’rob Am Hadrat Melech" – that we bring glory to Hashem by performing Misvot in large, public gatherings, and so there is certainly great value to having the Berit performed in a synagogue, as opposed to the couple’s small apartment. And, in many instances, it is not feasible for the couple to sleep with their baby in the synagogue on Friday night. Is there a way to permit bringing the infant to the synagogue on Shabbat for the Berit?
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) cites the Magen Abraham (Rav Abraham Gombiner, 1633-1683) as ruling that in a case where the Berit cannot, for whatever reason, be done in the home, a gentile may be asked to carry the baby to the synagogue through a Karmelit. Hacham Ovadia Yosef extended this ruling even to a case where the circumcision could be performed in the home, but it would be more respectable to have the Berit performed in a synagogue. The value of "Be’rob Am Hadrat Melech" allows having a gentile carrying the baby through a domain through which carrying is forbidden only "Mi’de’rabbanan." Hacham Ovadia further notes that even if the baby must be carried through a bona fide Reshut Ha’rabim (public domain), through which carrying is forbidden on the level of Torah law, there is a way to have the baby brought to the synagogue. According to some authorities (including the Rashba and the Ritba), carrying from a private domain to another private domain through a public domain without stopping is forbidden only "Mi’de’rabbanan." Therefore, it would be permissible to ask a non-Jew to pick up the baby at the house and then bring him directly to the synagogue without stopping at all in the middle. Since carrying in this manner is forbidden only "Mi’de’rabbanan" according to some opinions, and asking a gentile to perform a Melacha on Shabbat is forbidden only "Mi’de’rabbanan," this may be done for the sake of performing the Berit Mila in front of a large audience in the synagogue.
Hacham Ovadia Yosef adds that it would be preferable in such a case to ask a non-Jew to ask another non-Jew to carry the baby, as this makes the act of carrying even less directly linked to the Jew. Hacham Ovadia’s discussion of this topic appears both in his work Leviyat Hen, and in Hazon Ovadia – Shabbat (volume 3).
A separate question, which we will not discuss in this framework, is whether it would be similarly permissible to bring the infant back home after the Berit. The Mishna Berura writes that if the mother is home, and the baby needs to be with his mother in order to be nursed and cared for, then it would certainly be permissible to ask a non-Jew to bring the infant home. However, if the mother is also present at the Berit, and it would be difficult for them to remain in the synagogue the entire Shabbat, it is questionable whether there is room to allow the child to be carried home. This issue requires further discussion in a separate context.
Summary: Although a child is circumcised on Shabbat if it is his eighth day, it is forbidden to bring a knife through a public domain for the sake of facilitating the Berit Mila. If a larger crowd would be able to attend the Berit if it is held in the synagogue, as opposed to the baby’s home, and the baby needs to be carried through a public domain, a non-Jew may be asked to carry the infant. The non-Jew should be told to pick up the baby in the home and to carry him directly to the synagogue, without stopping at all in between. (The issue of whether the infant can then be brought back home after the Berit requires a separate discussion.)