There is a custom among Kabbalists to wear Tefillin while praying Minha, but the consensus among Kabbalistic scholars is that Tefillin should not be worn during Minha on Ereb Shabbat, particularly when one is praying Minha close to the onset of Shabbat. The reason for this exception is that as the sanctity of Shabbat is setting in, Tefillin are inappropriate. This view is cited by the Be’er Heteb commentary to the Shulhan Aruch (Orah Haim 37:3) in the name of the Arizal (Rav Yishak Luria of Safed, 1534-1572) and the Shelah (Rav Yeshaya Halevi Horowitz, 1558-1630). The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) writes (in Se’if Katan 100) that this was the custom of the Kabbalists in Yeshivat Bet-El. This is also the opinion of Rav Hida, Petah Hadebir, and Shelah Hakadosh. Although the Sefer Mekor Hayim says to wear Tefillin during Minha on Friday afternoon, Rabbi Avner Efgin in his Sefer Dibre Shalom volume 5 writes regarding that opinion and concludes that it is clear according to the Rashash that one should not wear Tefillin on Ereb Shabbat including the fast of Asara Be’Tebet.
Although the Shulhan Aruch rules explicitly that Tefillin may be worn on Ereb Shabbat until the period of Ben Ha’shemashot (the time between sunset and nightfall), the commentators explain that he is referring to somebody who had been wearing Tefillin since the morning. One should not, however, put on Tefillin in the afternoon on Ereb Shabbat.
This discussion becomes especially relevant when the fast of Asara Be’Tebet falls on Friday (as it does this year 2013, 5774). The custom in our community is to wear Tefillin during Minha on a fast day, and the question thus arises as to whether Tefillin should be worn when a fast is observed on Ereb Shabbat. Rav Haim Palachi (Turkey, 1788-1869), in his work Hikekeh Leb (vol. 1, Orah Haim 2), writes that Tefillin should not be worn during Minha on a fast day when it falls on Friday, just like on ordinary Fridays. This is the ruling followed by Hacham David Yosef, in his Halacha Berura (vol. 2, p. 190).
Summary: Although the custom in our community is to wear Tefillin during Minha on a fast day, if a fast day falls on Friday, Tefillin should not be worn during Minha.