Why do we not put on Tefillin on Shabbos?
Why is one not allowed to put on Tefillin on Shabbos?
And similarly, why is Tefillin Muktzah?
It is prohibited to put on tefillin on Shabbos and Yom tov, since tefillin are called אות (Shemos 13:9), and so is Shabbos (Shemos 31:17). Thus, donning tefillin is disrespect (זלזול) to the אות of Shabbos. (Shulchan Aruch, OC 31:1)
There is discussion amongst the poskim if, because tefillin are assur to wear, they should fall into the category of k'li shem'lachto l'issur (object designed for a prohibited use) on Shabbos. Mishna Berura (31:2) rules that one should consider them muktza, but may move them if necessary (בשעת הדחק), such as if they might fall on the ground or be stolen.
both tefillin and shabbos are refereed to as a sign between G-d and the Jewish people. we don't wear tefillin on shabbos b/c the day itself is that sign and wearing tefillin would take away from the sancityty of the day by doing something superfluous. Yom Tov as well is sometimes refered to as shabbos and while some of the restrictions regarding melacha are less strict for yom tov we apply the same reasoning to holidays as well. In regards to chol hamoed there is a dispute. Ashkenazim typically do while Sephardim and many Chassidim do not wear tefillin on chol haMoed. Muktsa is essentially not touching those things which may to be used on shabbos and since tefillin are not used on shabbos they are also muktsa
The Gemara in Menachos 36b says it's a gezeiras hakasuv and cites a machlokes Tannaim as to what pasuk it can be learned from:
R' Yosei HaGelili
דתניא (שמות יג, י) "ושמרת את החוקה הזאת למועדה מימים ימימה" ימים ולא לילות מימים ולא כל ימים פרט לשבתות וימים טובים דברי רבי יוסי הגלילי
“And you shall observe this ordinance in its season from year [miyamim] to year” (Exodus 13:10). This indicates that these mitzvot apply during the days [yamim] but not during the nights. Furthermore, the letter mem, meaning from, in the term: “From year [miyamim],” teaches: These mitzvot apply on some days, but not on all days. This excludes Shabbatot and Festivals, on which phylacteries are not worn.
דתניא ר' עקיבא אומר יכול יניח אדם תפילין בשבתות ובימים טובים ת"ל (שמות יג, טז) והיה לאות על ידך ולטוטפת בין עיניך מי שצריכין אות יצאו שבתות וימים טובים שהן גופן אות
Rabbi Akiva says: One might have thought that a person should don phylacteries on Shabbatot and Festivals. To counter this, the verse states: “And it shall be for a sign for you on your arm, and for a remembrance between your eyes" [This teaches that the obligation to don phylacteries applies when the Jewish people] require a sign [to assert their status as God’s nation, i.e., during the week. This serves to] exclude Shabbatot and Festivals, as they themselves are signs