The Shulchan Aruch paskens that we don't blow shofar on Rosh Hashana when it falls out on Shabbos (as it does this year). The reason the gemara gives is because we are afraid that one might carry the shofar in reshsus harabim and be oveir on the d'oreisa of hotza'ah. (This same reason applies for reading the megillah and taking the lulav). The question is, what happens if someone goes ahead and does blow shofar on Shabbos? Is he at least yotzei the mitzva d'oreisa of blowing shofar? It's understood that chazal made a gezeirah not blow shofar, but l'chorah the chiyuv d'oreisa still exists and if so why would one not be yotzei m'd'oreisa.
Before I get into the lomdus, there are actually several nafka mina to this question.
1) The Aishel Avraham in Siman 600 (and also brought in the otzar meforshim peirush on the side of the shulchan aruch in Siman 588 says that if one blew shofar on Shabbos Rosh Hashana then one does not make the Shehechiyanu on the second day. Since the beracha was made the first day there is no need to repeat it.
2) The Chachmas Shlomo (Siman 588) discusses a case where someone blew 2 set sof tekia teruah tekia but not that third set. This person is in limbo. .He was oveir on the d'rabanan of blowing shofar on Shabbos but he was not yet m'kayeim the d'oreisa since m'd'oreisa you need to blow 9 sounds-3 sets. The question is do we allow this person to blow 1 more set to be yotzei m'd'oreisa.
3) Chazal never made the takana in Yerushalayim in front of beis din. There is a machlokes rishinim what kind of beis din you need. The Rif paskins that all you need is a beis din of 3 people and you can blow in Yerushalayim. In 1881 (Rosh Hashana 5642) Rav Shlesinger wanted to blow shofar on Shabbos in Yerushalayim. In the sefer Ir Hakodesh V'HaMIkdash, Rav Tukitchinsky discusses the 2 sides of the issue. He writes that initaially Rav Shlesinger had 23 rabbonim on his side but in the end most of them backed out and only a few rabbonim blew shofar. He also quotes the gabbai of Rav Shmuel Salant who said that Rav Shmuel Salant would not pasken it is muttar but he wouldn't be mocheh against someone who went to hear the shofar. Fimally, Rav Tukitchinsky quotes the Aderes who said in 1903 that it was a shame Rav Shlesinger is not around to blow shofar on Shabbos since he would go behind a wall to hear him.
This leads us to the 3rd nafka mina. What would happen if someone blew shofar on Shabbos illegally-could one go hear the shofar and be m'kayeim a mitzva m'd'oreisa.
The lomdus is very simple. In fact Rav Elchanan in Kunteres Divrei Sofrim Siman 3 discusses this issue. When chazal were okeir a mitzva m'd'oreisa b'shev v'al ta'aseh were they okeir it completely and therefore even if you do the mitzvah m'd'oreisa you don't get a mitzvah or do we say that the mitzvah d'oreisa still exists. It's just that now I have 2 competing mitzvos. On the one hand I have the mitzvah to blow shofar. On the other hand I have the mitzvah to listen to the chachamim. Which mitzvah takes precedence? Rav Elchanan explains that we say "shev v'al ta'aseh", don't do anything. In this case it would mean don't blow the shofar.
Rav Elchanan has a few nafka mina and rayas from various gemaras for each tzad. ayin sham.
Rabbi Akiva Eiger seems to take a clear stand on this issue. In derush v'chiddush (ma'areches 8) he says that if you blow shofar on Shabbos you get a mitzva m'd'oreisa.
Rav Tzvi Pesach Frank (Har Tzvi Siman 88 and Mikroei Kodesh) asks that there is a Tosafos in Sukkah (3a) against Rabbi Akiva Eiger. If one sits ina sukkah and the table is in the house, you are not yotzei. Tosafos says you are not yotzei even m'd'oreisa because the rabanan have the ability to be okaeir the d'oreisa and you are not m'kayeim any mitzvah. The Ran disagrees with Tosafos and holds you are yotzei m'd'oreisa, but al kol panim, how can Rabbi Akiva Eiger go against a Tosafos. (Interestingly, Rav Elchanan never mentions this Tosafos). Rav Tzvi Pesach Frank wants to learn that the m'kor for Rabbi Akiva Eiger is from a different gemara and Tosafos is learning like a shitta that we don't pasken like (ayin sham).
I found in the journal HaPardes from 1960 that the Siridei Eish discusses this as well. He asks on Rav Tzvi Pesach Frank that we pasken like Tosafos. Therefore, whatever answer you give for Rabbi Akiva Eiger has to address Tosafos since that is how we pasken. He answers that there is a difference between sukkah and shofar. By the case of sukkah, chazal possuled the sukkah itself. In such a case we would say that you are not yotzei m'd'oreisa since you are not sitting in a sukkah. By shofar, they didn't possul the shofar but rather invalidated your act. However, your act of blowing teh shofar might not be good m'd'rabanan but it is good m'd'oriesa.
In Mikroei Kodesh, Rav Tzvi Pesach Frank does seem to mention this teretz but he doesn't stick with it.
Also, if you look in the Rishimus Shiurim on Sukkah, the Rav learns the Rambam not like the Siridei Aish. He is m'dayeik in the Rambam that it is not a p'sul in teh cheftzah shel sukkah but teh ma'aseh is not a mitzvah,