The general approach of the Gemara is that tefillin are a mitzvat aseh she-hazeman gerama – a positive time-bound commandment. When do we refrain from wearing tefillin?
Rabbi Yosei HaGelili points to the passage that commands that tefillin be worn mi-yamim yamimah – from day to day (see Sefer Shemot 13:10). He suggests that we can learn from here that it is only appropriate to wear tefillin during the day, and only on some of the days – excluding Shabbat and holidays.
Rabbi Akiva argues that the basic source to limit the commandment of tefillin to weekdays is the previous passage about tefillin “…and they shall be a sign (ot) on your hand and a remembrance between your eyes” (Shemot 13:9), which he understands to mean that tefillin are only necessary when there is a need for an ot – a sign. On days that are considered in and of themselves an ot, there is no need to don tefillin. There are many different explanations as to what makes Shabbat and Yom Tov days that are considered to be an ot. Some explain that the commandments regarding the holiness of these days make them a sign for the Jewish people. Some say that it is the fact that work is forbidden that makes such days stand out on the calendar as a sign. Yet others argue that it is the unique commandments of each of the days – sukka, matza, refraining from eating ḥametz, etc.
Some understand from this that it is prohibited to wear tefillin on Shabbat and holidays, since doing so is an act of belittling the unique ot of the holiness of the day. The Rashba even suggests that putting on tefillin on Shabbat or holidays involves the prohibition of bal tosif – that it is forbidden to add onto the commandments of the Torah. Nevertheless, the Rambam and the Tur rule that there is no prohibition, per se, although the Sages did forbid wearing tefillin in the public domain on those days.