Halacha requires reading special Haftarot on the three Shabbatot in between Shiba Asar Be’Tamuz and Tisha B’Ab, Haftarot that are called "Telata De’puranuta" – "the three of calamity." The first Haftara is taken from the beginning of Yirmiyahu (1:1-2:3), and for the second Haftara we read the subsequent prophecy of Yirmiyahu (2:4-28). On Shabbat Hazon, the Shabbat immediately preceding Tisha B’Ab, we read the first prophecy in the Book of Yeshayahu (1:1-27). All these prophecies foresee the destruction of the Bet Ha’mikdash and the Jewish people’s exile that resulted from their betrayal of God.
On the seven Shabbatot that immediately follow Tisha B’Ab, we read seven special Haftarot of consolation, prophecies from Yeshayahu that foresee Israel’s ultimate redemption. These Haftarot are known as the "Sheb D’nehamata" ("seven of consolation").
The Rishonim (Medieval Halachic authorities) debate the question of whether the special Haftara is read when Rosh Hodesh occurs on one of these Shabbatot. It occasionally happens that Rosh Hodesh Ab falls on the second Shabbat of the three weeks before Tisha B’Ab, and Rosh Hodesh Elul could fall on one of the Shabbatot during the seven weeks following Tisha B’Ab. Normally, when Rosh Hodesh falls on Shabbat we read a special Haftara for Shabbat Rosh Hodesh, the prophecy of "Ha’shamayim Kis’i" (Yeshayahu 66). The question thus arises as to which Haftara takes precedence when Rosh Hodesh falls on Shabbat during the three weeks before Tisha B’Ab, or during the seven weeks after Tisha B’Ab. Tosefot (Talmudic commentaries by French and German Medieval scholars) in Masechet Pesahim maintain that in such a case the standard Haftara for Shabbat Rosh Hodesh is read. Maran (author of the Shulhan Aruch), however, in his Bet Yosef (Orah Haim 425), cites the ruling of Rabbi David Abudarham (Spain, 14th century) that the special Haftarot of destruction and consolation are always read on their designated Shababtot, even on Rosh Hodesh. (Listen to audio recording for precise citation.)
This debate has yielded a difference of custom between the Ashkenazim and Sepharadim. Ashkenazim generally follow the ruling of the Rema (Rabbi Moshe Isserles, Poland, 1525-1572), who held that the Haftara for Shabbat Rosh Hodesh is read in this case. Sepharadim, however, follow the position of the Shulhan Aruch (Orah Haim 422:1), who ruled that when Rosh Hodesh Elul falls on Shabbat the Haftara of consolation is read instead of the Haftara for Shabbat Rosh Hodesh. Although he does not specifically address the other case, when Rosh Hodesh Ab falls on Shabbat, it stands to reason that regarding that situation, too, he follows the Abudarham’s ruling, requiring that the Haftara for the three weeks be read. This is the ruling of Hacham Ovadia Yosef and of the work Nehar Misrayim. This is, indeed, the custom we follow.
One might, at first glance, wonder why we would not apply in this case the famous principle of "Tadir Ve’she’eno Tadir Tadir Kodem," which affords priority to the more frequent Misva. Since Rosh Hodesh occurs more frequently than the Shabbatot before and after Tisha B’Ab, perhaps the special Haftara for Rosh Hodesh should supersede the Haftarot of destruction and consolation. The response to this argument is that the rule of "Tadir" does not take effect when the Sages specifically ordained performing a certain Misva at a certain time. Once Hazal enacted this provision to read special Haftarot on these Shabbatot, we read these Haftarot even if other factors would have dictated reading a different Haftara. Furthermore, the Maharil (Rabbi Yaakov Molin, Germany, 1365-1427) noted that the Haftara for the second of the three weeks actually contains a thinly-veiled reference to Rosh Hodesh Ab: "All who seek her are not weary; they find her in her month" (Yirmiyahu 2:24). This verse speaks of the enemies of Israel who defeated her in the month of Ab. Hence, this Haftara bears relevance not only for the three weeks, but also specifically to the day of Rosh Hodesh Ab.
In any event, each community should follow its practice in this regard.
Summary: Halacha requires reciting special Haftarot on the Shabbatot during the three weeks before Tisha B’Ab, and during the seven weeks after Tisha B’Ab. The common practice among Sepharadim is to read these Haftarot even in a case where Rosh Hodesh falls on Shabbat, whereas Ashekenazim generally afford preference to the special Haftara for Shabbat Rosh Hodesh in such a case.
