It is common knowledge that we do not sound the shofar on Rosh HaShana that coincides with Shabbat. The first Mishna in the fourth perek teaches that this is only the case outside of the Mikdash. In the Temple the shofar was blown even on Shabbat. Following the destruction of the Temple, Rabban Yohanan ben Zakkai established a rule whereby the shofar was blown on Shabbat in places that had an established bet din. Rabbi Elazar argues with that tradition and says that Rabban Yohanan ben Zakkai established that rule only in Yavne.
In searching for an explanation why the shofar is not sounded on Shabbat, our Gemara rejects the suggestion that this is a Biblical law based on the difference between the pasuk in Bamidbar (29:1), which calls Rosh HaShana a day of teru’ah (i.e. blowing the shofar), and the pesuk in Vayikra (23:24) that refers to it as a day of zikhron teru’ah (when we remember the blowing of the shofar), the former referring to a regular year and the latter to Rosh HaShana falling on Shabbat. Instead, our Gemara suggests that it is a Rabbinic ordinance, quoting Rabbah as teaching that our concern lest the shofar be carried in a public place – which is forbidden on Shabbat – forced the Sages to suspend the mitzva. In the Temple – or subsequently in the time of Rabban Yohanan ben Zakkai in Yavne – where mitzvot were carried out with great care, there was no need for this Rabbinic ordinance, and blowing the shofar on Shabbat was permitted.
It is interesting to note that the Talmud Yerushalmi accepts the argument that it is a Biblical commandment to refrain from blowing the shofar on Shabbat.