Commentary for Sukkah 4:2
לוּלָב שִׁבְעָה כֵּיצַד, יוֹם טוֹב הָרִאשׁוֹן שֶׁל חָג שֶׁחָל לִהְיוֹת בְּשַׁבָּת, לוּלָב שִׁבְעָה, וּשְׁאָר כָּל הַיָּמִים, שִׁשָּׁה:
Lulav — seven. How so? If the first day of the festival fell out on Shabbath, lulav — seven. [For the first day (of the festival) overrides Shabbath, it having a basis in the Torah (for taking the lulav even) in the borders (i.e., outside of the Temple), viz. (Leviticus 23:40): "And you shall take for yourselves on the first day"]. All the other days, six. [If the first day of the festival falls out on any of the weekdays, so that Shabbath falls out on Chol Hamoed, it does not override. And even though all seven days obtain in the Temple by Torah law, the sages established their decree (against taking the lulav) for the other days of the festival, since they have no Torah basis (for taking the lulav) in the borders (on those days). And, in truth, even today the taking of the lulav should override Shabbath on the first day of the festival; but because we are not expert in the fixing of the New Moon, so that it (that day) may not be the first day of the festival, we do not override the Sabbath on the basis of a possibility. And in taking it, there is no (Torah proscribed) Sabbath labor, but merely the moving of it. (Still, it is forbidden) lest he take it in his hand and go to an expert to learn the blessing or the order of the shakings (na'anuim) and come to carry it four cubits in the public domain. And this is also the reason that we do not blow the shofar or read the Megillah (on Shabbath)].
Bartenura on Mishnah Sukkah
English Explanation of Mishnah Sukkah
As we have learned before, outside of Jerusalem it is a mitzvah from the Torah to take the lulav only on the first day of the festival. Hence, if the first day of the festival falls on Shabbat, one still performs the mitzvah of the lulav. We saw this described above in 3:13. In such a case the lulav will be taken for seven days one day in which the mitzvah is “deoraita” from the Torah, and the rest of the days it is “derabbanan” from the rabbis.
However, if Shabbat falls on another day besides the first day of the festival, then since the mitzvah is only of rabbinic origin, it is not observed. In other words, the derabanan mitzvah of the lulav does not override the Shabbat. In such a case the mitzvah is observed for only six days.
Since the time of the Babylonian Talmud it has become customary not to take the lulav on Shabbat, even if it falls on the first day of the festival. However, this is not the custom reflected in the mishnah.