The Shulchan Arukh records that although a Torah scroll is circled on the bimah each day of Sukkot and seven times on Hoshana Rabbah, on Shabbat neither the circling nor the removal of scrolls onto the bimah is done at all (Shulchan Arukh, Orach Chayim 660:1), a ruling echoed by the Tur (Orach Chayim 660) and the Arukh HaShulchan (Orach Chaim 660:3).
The Mishnah Berurah and Kaf HaChaim explain that the reason hakafot are omitted on Shabbat is that they correspond to the taking of the lulav, and just as the lulav is not taken on Shabbat, so too the hakafot commemorating the Temple service are not performed, and consequently the Torah scrolls are not even brought out to the bimah (Mishnah Berurah 660:4; Kaf HaChayim on Shulchan Arukh, Orach Chayim 660:10).
This is rooted in the Gemara's discussion of why the lulav, unlike the willow branch, is not taken on Shabbat even when Sukkot falls on Shabbat, out of concern that a person might carry it through a public domain to learn how to wave it properly (Sukkah 43b:3), a distinction further analyzed by Sukkah 44a:1 and Tosafot on Sukkah 43b:3.
The Beit Yosef preserves the parallel line of reasoning that just as one lacking a lulav or etrog does not circle the bimah on weekdays, Shabbat itself proves the point, since the lulav is not taken and the hakafot are not performed (Beit Yosef, Orach Chayim 660:6).
A comparable pattern appears regarding the shofar: the Shulchan Arukh and its later commentators state that when Rosh Hashanah falls on Shabbat, the shofar is not blown, for fear that someone might carry it to an expert to learn the blowing (Shulchan Arukh, Orach Chayim 588:5; Levush, Orach Chayim 588:5; Peninei Halakhah, Days of Awe 3:16), a rationale also detailed in Shulchan Arukh HaRav, Orach Chayim 588:4.
The Divrei Yatziv considers a reported custom of circling the table seven times on Friday night and, finding no precedent for it, suggests it may stem from extending the hakafot of Kabbalat Shabbat, concluding that where Shabbat is accepted only later such hakafot should not be practiced since they are merely completing an earlier tikkun (Divrei Yatziv Likutim 28).
Igros Moshe addresses a related question of why public trumpet blasts to gather the people for hakhel were not permitted on Shabbat, weighing whether this stemmed from hakhel not being a Temple service, from the possibility of postponement, or from the gathering itself not warranting an override of Shabbat restrictions (Igros Moshe YD 4:65).