The Mishna in Rosh Hashana (29b) states that they blew shofar in the mikdash but not בגבולין (for our purposes outside of Yerushalayim, there is a big machlokes rishonim what it means exactly). The Gemara asks why don't we blow on Shabbos? The Gemara first brings R' Levi Bar Chama that it is a דין דאורייתא based on pesukim, the Torah says in one place יום תרועה and in another place זכרון תרועה. One is on a weekday one is on Shabbos. The Gemara rejects this and says it is because of Rabbah's gezerah that he will carry. Interestingly enough the Yerushalmi does not reject the hava amina and the simple understanding of the Yerushalmi is that not blowing shofar on Shabbos is a דין דאורייתא. What is even more interesting is that when Rosh Hashana does fall out on Shabbos, in davening we say זכרון תרועה.
Tosafos asks why is it that after the churban habayis the חכמים allowed blowing the shofar on Shabbos in a place where there is a בית דין, while by לולב after the churban we simply don't take it on Shabbos even where there is a בית דין? They answer that the shofar's purpose is to bring our זכרונות up to שמים and therefore Chazal felt that they could not completely abolish the practice of blowing shofar on Shabbos, there had to be somewhere where the shofar was blown. Based on this, what about today? We don't blow shofar anywhere when Rosh Hashana falls out on Shabbos.
As I mentioned above the Gemara explains that we don't blow on Shabbos because of גזירה דרבה. However, it is clear from the Mishna that this was a much earlier גזירה as רבה is one of the late amoraim and the Mishna states that in the time of the Beis Hamikdash they did not blow shofar on Shabbos outside of Yerushalayim. We see that the גזירה was in existence for a long time before רבה, it is a little strange that the Gemara attributes it to רבה.