When Rosh Hashanah falls on a Shabbos, we don't blow the shofar. There are several reasons for this law. I'd like to present three. The Talmud (Rosh Hashanah 29b) cites the teaching of Rava that we are concerned lest a Jew somewhere violates the Shabbos by carrying the shofar in a public domain (r'shus harabim). The rabbis issued a decree that it's better to remain passive and abstain from the biblical mitzvah of shofar, to protect the integrity of Shabbos. This halacha speaks volumes about the halachos of Shabbos in general and specifically the laws of carrying. This is especially significant for many of us who are blessed with an eruv. We must educate our children regarding the laws of carrying and their need to inquire of the existence of an eruv when they go away for Shabbos.
The Meshech Chachma (Vayikra 23:24) provides another reason. He begins by extolling the significance and importance of shofar. The Talmud (Rosh Hashanah 26a) equates the blowing of the shofar to a service that is performed in the Holy of Holies in the Temple; not many other mitzvos are awarded such a distinction. Moreover, our rabbis understand the blowing of the shofar as an antidote to the harsh charges of the Satan. The shofar, as a medicine, has the ability to combat the many different strands of virus hurled against us. Thus, not blowing the shofar is akin to a sick patient not taking his medicine. Whatever good excuse the patient has does not compensate for the absence of the medicine.
It is precisely because the shofar is so crucial and important to the welfare of the Jewish People that we abstain from blowing the shofar on Shabbos. The Jewish Community is saying to Hashem, "Your Torah and Your Shabbos is even more important and dear to us than our individual and collective needs". Similarly, the Talmud (Berachos 6a) teaches that Hashem dons tefillin as do the Jewish people; our tefillin contain the shema in praise of Hashem, while his tefillin contain the verse "and who is like Israel, a unique nation in the land," in praise of Israel. It is our sacrifice of foregoing the benefits of the shofar on behalf of the benefit of Shabbos that serves in our stead. Our abstaining from blowing the shofar on Shabbos is placing Hashem before ourselves.
The Chassam Sofer offers yet another, most fascinating, reason for not blowing the shofar on Shabbos. He explains that the nature of Avraham's command to do the akieda was very different than Yitzchak's. Avraham heard the command directly from Hashem, which is to be likened to a command in the Torah Sheb'ksav - the Written Torah, which emanated directly from Hashem. Yitzchak heard the will of Hashem through his father, which he compares to the Torah Sheb'al Peh - the Oral Law. Avraham was ready to offer his son based upon G-d's direct instruction, while Yitzchak demonstrated his willingness to offer his life on behalf of the Torah Sheb'al Peh.
It is for this reason that Yitzchak's sacrifice was even greater than that of his father's. Therefore, the Chassam Sofer explains, we refer in our Mussaf service to this event as the binding of Yitzchak - "v'akeidas Yitzchak l'zaroh tizkor - and may the merit of the binding of Yitzchak be remembered on behalf of his progeny" - as opposed to the binding of Yitzchak by Avraham, as Yitzchak proved himself ready to sacrifice his life based on the Torah Sheb'al Peh. In this regard, his sacrifice was greater than his father's. Avraham was convinced that he head heard G-d speaking to him, but Yitzchak who did not hear it from Hashem demonstrated his absolute trust and faith in the masorah - the oral tradition.
In addition, the Torah (Breishis 22:6) describes the journey of father and son as "vayelcho shneihem yachdav". This refers to the integration and synthesis of the written and oral law.
The Talmud (Rosh Hashanah 16a) provides as a reason for the blowing of the shofar, "Hashem said 'blow the ram's horn before me so that I shall recall on your behalf akeidas Yitzchak'". That is, to recall the sacrifice of Ytitzchak on behalf of Torah Sheb'al Peh. Therefore when Rosh Hashanah falls on a Shabbos our abstaining from blowing the shofar based upon Torah Sheb'al Peh is a living implementation of a teaching of the akeidah. As Yitzchak demonstrated his absolute allegiance and commitment to the masorah, our refraining from blowing the shofar on Shabbos is our affirmation of the masorah as well.