One does not wear Tefillin on Shabbat or Yom Tov. See Shulchan Aruch 31:2, and Rama. This includes both the first day of Yom Tov and the additional day observed in the Diaspora. The custom among Sepharadim is not to wear Tefillin during Chol Ha'mo'ed Pesach or Chol Ha'mo'ed Sukkot, either. Among Ashkenazim, different customs exists in this regard of Chol HaMoed. Some wear Tefillin with a Beracha, some wear them without a Beracha, and some do not wear them at all. Tefillin are worn on Chanukah and Purim. On Purim, there is a custom to kiss the Tefillin during the reading of the Megila, when the reader reaches the word "Ve'ykar" in the verse, "La'Yehudim Hayeta Ora Ve'simcha Ve'sason Ve'ykar" ("The Jews had light, joy, jubilation and glory"), because the word "Ve'ykar" alludes to the Mitzva of Tefillin. (Ben Ish Chai, Hilchot Purim Halacha 9)
Given that Tefillin may not be worn on Shabbat or Yom Tov, the question arises as to whether they are considered Muktze – an object that serves no functional purpose on Shabbat and may therefore not be handled on Shabbat. For example, if one wants to sit on a chair on Shabbat but his Tefillin were left on the chair before Shabbat, may he move the Tefillin so he can sit? Furthermore, if a Rabbi teaches a class on Shabbat about Tefillin, may he show the students Tefillin as part of his instruction?
Halacha classifies Tefillin under the category known as "Keli She'melachto Le'issur" – utensils whose primary use involves activity forbidden on Shabbat. The law regarding such a "utensil" is that one may move it either to use the space it currently occupies, or to use the item itself for a permissible purpose. Therefore, in the aforementioned case of the Tefillin left on a chair, one may, indeed, move the Tefillin so that he can use the chair. Likewise, one may use Tefillin for a permissible purpose on Shabbat, such as if a Rabbi wishes to show the Tefillin to his students as part of his instruction of the laws of Tefillin.
Summary:
1) One may not wear Tefillin on Shabbat or Yom Tov; different customs exist regarding wearing Tefillin on Chol Ha'mo'ed.
2) Tefillin are worn on Chanukah and Purim.
3) On Shabbat and Yom Tov, one may move Tefillin only for the purpose of using the space it occupies, or to use the Tefillin themselves in a manner permissible on Shabbat.
See Halichot Olam, Helek 3, page 196. See Menuhat Ahava, Helek 1, page 263.
