Hacham Ovadia Yosef was asked whether it would be permissible for a Mohel to walk on Shabbat (or Yom Tob) beyond the point where walking is allowed on Shabbat, in order to perform a Berit. The prohibition of "Tehum Shabbat" forbids walking beyond a certain distance (2000 Amot) outside one’s town on Shabbat, and the question thus arises as to whether this prohibition is overridden in the case of a Berit which is to be performed on Shabbat. In some rural regions, where people live in small towns which are situated at a distance from one another, it might be necessary for a Mohel to walk beyond the "Tehum Shabbat" limit on Shabbat in order to perform a Berit in a different town. Would this be allowed?
Hacham Ovadia cites a discussion in the Talmud Yerushalmi (Rosh Hashanah 2:3) from which it emerges that walking beyond "Tehum Shabbat" is allowed for a Berit Mila on Shabbat only according to the minority position of Rabbi Eliezer. Rabbi Eliezer maintained that just as the act of Berit Mila overrides the Shabbat prohibitions on a child’s eighth day, similarly, all "Machshireh Mila" – preparations that need to be done for the purpose of a Berit – also override the Shabbat prohibitions. According to Rabbi Eliezer, the Yerushalmi indicates, a Mohel would be allowed to walk beyond the "Tehum Shabbat" limit on Shabbat in order to perform a Berit. The clear implication of the Gemara is that according to the accepted view, that preparations for a Berit do not override the Shabbat prohibitions, this would not be allowed. And thus Hacham Ovadia Yosef concludes that it is forbidden for a Mohel on Shabbat or Yom Tob to walk beyond the "Tahum Shabbat" in order to perform a Berit. Hacham Ovadia notes that this was the position taken by Rav Tzvi Pesach Frank (1873-1960), in his work Har Sevi.
However, Hacham Ovadia makes an exception in the case of a Berit that is to be performed on Yom Tob Sheni Shel Galuyot – the additional day of Yom Tob observed in the Diaspora. He notes that this day of Yom Tob is observed only Mi’de’rabanan – on the level of Rabbinic enactment – and according to some authorities, the prohibition of "Tehum Shabbat" is also Rabbinic in origin. Therefore, given the lower level of stringency of "Tehum Shabbat" on Yom Tob Sheni, Hacham Ovadia allows a Mohel to walk beyond "Tehum Shabbat" on Yom Tob Sheni to perform a Berit. (Hacham Ovadia adds that this applies even to walking a distance of 12 "Mil," which is generally treated more stringently, since some opinions maintain that walking even this distance on Shabbat or Yom Tob is forbidden only "Mi’de’rabanan.")
Summary: Although the Misva of Berit Mila on a child’s eighth day overrides Shabbat, it is forbidden for a Mohel to walk beyond the "Tehum Shabbat" outside his town for the purpose of performing a Berit on Shabbat or Yom Tob, though this is allowed on the second day of Yom Tob added in the Diaspora.