As promised at the end of this post.
Brachos 47b - ...Once Rebi Eliezer entered the beis knesses and did not find ten men [for a minyan] and he freed his slave to complete the minyan...How could he do this, for Rav Yehuda said, 'whomever frees his slave transgresses the positive commandment of "leolam bahem taavodu?'" For a mitzvah, it is different. But this is a mitzvah which comes about through an aveira? For a public mitzvah, it is permitted.
Rashi (s.v. mitzvah) says the public mitzvah is Kedusha, which the public couldn't fulfill without a minyan. To him, Kedusha is a mitzvah dioraissa (probably a kiyum of venikdashti, but not a chiyuv - the Chinuch 296 does not even mention this as part of the mitzvah of venikdashti. See, however, Torah Shleima Mikeitz (Breishis 42:5) 31 end - a befuddling matter is discussed in Shut Rashbatz Chelek II 163 someone who took a shvua not to be chazzan, the shvua does not take effect. He says, "It appears to be a mitzvas asei from the Torah, for to be chazzan is to be motzei other people their obligation for dvarim shebikdusha and kriyas haTorah, and both are from the Torah, because devarim shebikdusha they said...venikdashti...if so, someone who vows not to be chazzan is like vowing not to make a sukkah." Torah Shleima continues, we may note from the Maharil in hilchos Eruvei Chatzeiros (p. lamed alef amud alef) "Mahari Segal said he didn't see any posek allow one to make an eruv tchumin to go to beis knesses except the Smak and the sefer Agudah." It seems these rishonim hold it's even a chiyuv midioraissa, whereas Rashi and many Rishonim hold Borchu & Kedusha is only a kiyum of venikdashti.
(About the Rashbatz comparing this vow to a vow not to build a sukkah - but building a sukkah isn't a mitzvah! See Haamek She'eilah 169:1.)
Tosfos (s.v. mitzvah) aska, this episode is a question to the Behag who says one observes aveilus on the last (eight) day of Yom Tov (which is only midirabanan), and the mitzvah diribanan of the last day of Yom Tov does not push off the mitzvah dioraissa of aveilus...as we see here, Tefila dirabanan pushes off the dioraissa of leolam bahem taavodu....
Tosfos seems to hold either Kedusha is not included in the mitzvah of "venikdashti besoch bnei yisrael" or there was tefila betzibur without kedusha (maybe maariv, but highly unlikely. His shita of arvis (Maariv) reshus is [Brachos 26a s.v. ta'ah] that if another mitzvah comes at the same time as Maariv, you do the other mitzvah first. It is unlikely that Rebi Eliezer would free his slave for maariv which is only reshus.) Bottom line, Tosfos holds the mitzvah dirabanan outweighs the asei dioraissa of not freeing your slave.
The Rosh (siman 20), however, is more similar to Rashi: he freed his slave "for kedusha and Borchu, even though they have no source in the Torah. For it does not seem to me to be discussing a minyan for a dioraissa matter like to read Parshas Zachor...." Most Rishonim do not agree with this Rosh, for mentioning Amalek without reading parshas Zachor is enough. See Chinuch 603. Minchas Chinuch says explicitly: "to read Parshas Zachor out of a Torah and with a minyan is midirabanan, but midioraissa it's enough for each individual to remember it verbally, and this is simple." But the Rosh obviously holds there is a midioraissa requirement to read parshas Zachor with a minyan.