Gemara

(Beraisa - R. Yosi): Burning [on Shabbos] was taught by itself to teach that it is only a Lav [without Misah or Kares];

R. Noson says, it was taught by itself to be Mechalek. (Just like it is an Av Melachah, and one is liable for it by itself, the same applies to all Avos.)

Rosh Hashanah 16b (R. Yitzchak): Any year in which they did not blow the Shofar at the beginning, they will blow it (or bad things will occur) at the end, because the Satan was not confused.

29b (R. Levi bar Lachma) Contradiction: It says "Shabboson Zichron Teru'ah", and it says "Yom Teru'ah"!

Answer: When Yom Tov is on Shabbos [it is merely a commemoration. We do not blow the Shofar]. When Yom Tov is a weekday, we blow.

Yevamos 20a (Mishnah): A widow to a Kohen Gadol [may not do Yibum in any case].

Question: Granted, if she was widowed from Nisu'in, a Lav and an Aseh forbid her, and an Aseh (Yibum) does not override them.

However, if she was widowed from Kidushin, only a Lav forbids her. The Aseh should override it!

20b - Answer (Rava): We forbid Yibum lest they have a second Bi'ah [which is forbidden].

Support (Beraisa): If they did Yibum, the first Bi'ah acquires her. A second Bi'ah is forbidden.

Rishonim

Rosh (Hilchos Tzitzis (after Menachos) 17): Chachamim decreed not to put Tzitzis, even of linen, on a linen garment. R. Tam says that if one wears a linen Talis and blesses on it, it is a Berachah l'Vatalah.

Talmidei R. Yonah (Berachos 1a DH v'Chachamim): Chachamim who argue with R. Gamliel say that one who read Keri'as Shema after midnight was not Yotzei even b'Di'eved. Would they hold that one should not read then?! The Torah obligates reading. How can Chachamim exempt him from this? Some say that after midnight he reads without the Berachos, for they are only mid'Rabanan, but the Torah Mitzvah applies the entire night. R. Yonah holds that Chachamim say that he does not read at all after midnight. Chachamim can exempt from an Aseh, for the sake of a fence, or due to fulfilling the Mitzvah itself, like we find regarding Lulav. The Torah Mitzvah is only the first day, and when it is Shabbos, Chachamim exempted him from taking it, lest one carry it four Amos in Reshus ha'Rabim. The Torah obligated a linen garment in Tzitzis and permitted Sha'atnez, and Chachamim exempted it. Likewise, they totally exempted from Keri'as Shema.

Tosfos (Yevamos 90b DH Kulhu): The Gemara said that Chachamim can uproot Torah laws only through commanding to be passive. How did they uproot a linen garment from Tzitzis? One transgresses through an action, i.e. putting it on! Really, he is not obligated until it is on him. Once it is on him, he is passive. (h/e does not attach Tzitzis/)

Tosfos (16b DH she'Ein, citing Bahag): 'They did not blow the Shofar' discusses when Ones prevented them, but not if Rosh Hashanah fell on Shabbos.

Tosfos (Pesachim 5b DH Lechalek): The Riva says that according to the opinion that the Torah explicitly forbids burning on Shabbos to teach that it is [only] a Lav, it is not forbidden on Yom Tov, for it is not called a Melachah.

Poskim

Shulchan Aruch (OC 588:5): When Rosh Hashanah is on Shabbos, we do not blow the Shofar.

Chachmas Shlomo (5): If one transgressed or forgot and blew the Shofar twice (with a Teki'ah before and after each time) on Rosh Hashanah that is Shabbos, should he blow a third time to fulfill the Torah Mitzvah? He already transgressed Chachamim's decree. If he blows again, at least he did the Torah Mitzvah. If not, he transgressed without doing a Mitzvah! Perhaps in such a case Chachamim did not enforce their decree to uproot a Mitzvah, since in any case he transgressed. I learn from Beitzah 21a. If the lambs (brought with Shtei ha'Lechem) were slaughtered Lo Lishmah or before the time, if it was Shabbos, we do not throw the blood. If one threw it, it is Kosher to permit burning the Eimurim at night. The Gemara infers that l'Chatchilah one may not throw the blood, even though this causes that the Shechitah was an Isur Torah, and if he would throw the blood, the Korban is Kosher. However, there, even if he throws the blood, he transgressed mid'Oraisa. Also, it is not an obligation of the day, since he slaughtered them Lo Lishmah or too early. Here, there is a Torah Mitzvah to blow the Shofar today. If he blew he was Yotzei, just he transgressed a mid'Rabanan law. This requires investigation.

R. Akiva Eiger (Drush v'Chidush Ma'areches 8, DH Ulam Leshon): If one blew the Shofar on Rosh Hashanah that is Shabbos, he fulfilled the Mitzvah. It applies also on Shabbos. However, he transgressed an Isur mid'Rabanan of Shabbos.

Avnei Nezer (YD 141:11): Tosfos holds that Chachamim uproot only through Shev v'Al Ta'aseh (commanding to be passive) , and the Mitzvah is uprooted automatically. If so, they did not uproot the Mitzvah of Teki'as Shofar when Rosh Hashanah is on Shabbos. This would be an overt uprooting! Rather, the Mitzvah remains in its place. Yevamos 20b proves this. There is no Mitzvah of Yibum if a Lav and an Aseh forbid the Yevamah. If only a Lav forbids her, mid'Rabanan they may not do Yibum. if they did Yibum, he acquires her. This shows that the Mitzvah of Yibum remains in its place. If one did it, the Aseh was Docheh the Lav, even though Chachamim enacted not to do it. They merely permitted to be passive, and automatically the Mitzvah is Batel. Do not say that Chachamim uprooted the Mitzvah. They do not have the power to do so!

Minchas Yitzchak (2:50): If one blessed and blew the Shofar on Rosh Hashanah that is Shabbos, was he Yotzei b'Di'eved? If he was, he does not bless Shehecheyanu on Yom Tov Sheni according to one opinion. It was obvious to R. Akiva Eiger that he was Yotzei. Also Avnei Nezer says so. However, Chelkas Yo'av (Kava d'Kashyasa 99) brought many proofs that when Chachamim exempted from a Mitzvah, they totally uprooted it. They have the power to totally uproot an Aseh through Shev v'Al Ta'aseh.

Mishnah Berurah (600:7): If a Tzibur did not get a Shofar until late on the second day of Rosh Hashanah, which was Erev Shabbos, and they already prayed Ma'ariv of Shabbos, but it was still day, and the only one who knows how to blow already accepted Shabbos, he should blow only three sets (Shevarim-Teru'ah, Shevarim, and Teru'ah, each with a Teki'ah before and after it), and no more. If it is already Bein ha'Shemashos, he should not blow at all, for the Chiyuv to blow is only mid'Rabanan. However, there is a Torah Chiyuv to blow on the first day, so then he must blow without a Berachah.

Kava d'Kashyasa (cited in Minchas Yitzchak): Surely, Chachamim totally uprooted the Mitzvah. If not, nowadays that we do not have Reshus ha'Rabim, we should blow the Shofar and take the Lulav, just like nowadays we permit ornaments [because there is no Reshus ha'Rabim]. This is not a solid proof, for there they immediately permitted ornaments in a Chatzer. Shofar was forbidden even in the house. However, this itself proves that it was totally exempted. Also, if the Mitzvah were not uprooted, we should blow Bein ha'Shemashos, for then Shevusim (mid'Rabanan Isurim of Shabbos) do not apply, even if we can say 'either way you say!' (I.e. if it is day, it is forbidden on Shabbos. If it is night, there is no Mitzvah, so it is forbidden mid'Rabanan on Yom Tov - PF.)

Minchas Yitzchak (2): Many Meforshim say that even nowadays many cities including London are Reshus ha'Rabim. Every day 600,000 people go there. The Mishnah Berurah (600:7) supports what Kava d'Kashyasa said about Bein ha'Shemashos. Why does it matter whether they did not blow due to Ones, or because Rosh Hashanah fell on Shabbos? Also on Shabbos, they are obligated to blow, just they are Anusim due to Chachamim's enactment! Rather, this shows that Chachamim's enactment exempts. Another Ones does not exempt. One who was Ones is not like one who fulfilled (Rambam Hilchos Korban Pesach 5:2, Kli Chemdah Parshah Tetzei 4).

Minchas Yitzchak: The Yerushalmi says that in Yerushalayim they blew mid'Oraisa (on Shabbos), and in Yavneh they blew only mid'Rabanan [after R. Yochanan ben Zakai enacted to blow where there is Beis Din]. If Chachamim were not Mevatel the Mitzvah on Shabbos, and b'Di'eved one fulfilled the Mitzvah, they would fulfill the Torah Mitzvah [also in Yavneh]! Rather, they were Mevatel the Mitzvah totally. Really, this is not a proof. The Yerushalmi is like the opinion that learns from a verse to exempt mid'Oraisa. Surely R. Yochanan ben Zakai's decree is only mid'Rabanan. The question was according to the opinion that Teki'as Shofar is merely a decree mid'Rabanan.

Pnei Yehoshua (Rosh Hashanah 30a DH b'Tosfos): The Beraisa says that Teki'os of Rosh Hashanah bi'Gevulin (outside the Mikdash/Yerushalayim) override Shabbos. After people heard in Beis Din, they blew again in their houses for the ideal Mitzvah, or to be Motzi their households. In the house, there is no need to decree lest one carry in Reshus ha'Rabim. A Toras Kohanim teaches that the Teki'os of Yom Kipur [in Yovel] are Docheh Shabbos, but those of Rosh Hashanah are Docheh only in Beis Din. Rashi holds that the Beraisa forbids only due to carrying, but Toras Kohanim holds that Teki'ah is a quasi Melachah. Only Meleches Avodah is forbidden on Yom Tov. It is like an Isur Aseh, like the Aseh that one's animal rest on Shabbos, and according to Beis Shamai that one's Kelim rest. Since there is no Kares or Misah, the Aseh of Teki'as Shofar overrides it. It is forbidden bi'Gevulin, since they are unsure which day is Rosh Hashanah.

Minchas Yitzchak: If Beis Din uprooted the Mitzvah, how could R. Yochanan ben Zakai enact to blow where there is Beis Din? The Turei Even similarly asked regarding the Gemara's connotation that R. Levi holds that it is a proper Melachah, how Chachamim were able to enact extra Teki'os. Mitzpah Eisan answered that R. Levi distinguishes between Shabbos and Yom Tov. It is a mere Isur, but not a proper Melachah, so it is permitted on Yom Tov, like Tosfos said according to the opinion that burning [on Shabbos] is only a Lav. This does not answer for R. Yochanan ben Zakai's enactment on Shabbos. However, if we say that the Mitzvah is not Batel, the Aseh overrides the Melachah of Teki'ah. However, if R. Levi expounds that it does not apply on Shabbos, there is no Aseh then! Perhaps he expounds that we may blow in Beis Din, like the Yerushalmi expounds a Heter to blow in Yerushalayim. Rashi (Vayikra 25:9) brings such a Drashah from Toras Kohanim, that the Shofar of Rosh Hashanah overrides Shabbos only in Beis Din. R. Yochanan ben Zakai 'enactment' was really this Drashah. The Stam Bavli is unlike R. Levi. However, the Yerushalmi says that R. Yochanan ben Zakai's enactment was mid'Rabanan.

Mishnas R. Aharon (2 p.201): When Yisrael blow more Teki'os than the Torah requires, the Satan sees that Hash-m should show favoritism to them. This confounds him. When Rosh Hashanah is on Shabbos and we forfeit this great advocate, out of great Yir'as Shamayim, lest someone transgress carrying, this similarly confounds him!