Gemara

46a (Beraisa - R. Eliezer ben Yakov): I heard that Beis Din can lash or punish not according to Torah law. They do not intend to change Torah law, rather, to make a fence (to deter transgressors).

Once, a man had Bi'ah with his wife under a fig tree, and Beis Din lashed him. He did not deserve to be lashed, but at the time it was necessary.

52b: A Bas Kohen was Mezanah. Rav Chama bar Tuvya burned her in a fire of sticks.

Rav Yosef: He made two mistakes. He erred about Rav Masnah's teaching (that the Pesilah for Sereifah is molten lead), and about a Beraisa:

(Beraisa): "U'Vosa El ha'Kohanim... v'El ha'Shofet Asher Yihyeh ba'Yomim ha'Hem" - we judge capital cases only at a time when there are (Shoftim, i.e. the Sanhedrin in Lishkas ha'Gazis, near the) Kohanim.

Kidushin 81a (Rav): We lash someone about whom there are constant rumors of immorality - "Al Beni Ki Lo Tovah ha'Shemu'ah" (cease my sons, for there are bad reports).

Rishonim

The Rif and Rosh (Sanhedrin 6:1) bring the Gemara (Sanhedrin 46a) verbatim.

Question: Why does it say that Rav Chama erred about judging capital cases when there is no (Sanhedrin near the) Kohen? Beis Din can punish not according to Torah law!

Answer #1 (Nimukei Yosef 16a DH Garsinan): Only the Great Sanhedrin is authorized to do so. Even though Rav Huna cut off a man's hand (Sanhedrin 58b), and the Reish Galusa was prepared to gouge out the eyes of a murderer (Sanhedrin 27a), this is to give some of the Torah punishment. Only the Great Sanhedrin can give the full punishment.

Rebuttal (and Answer #2 - Hagahos ha'Bach 3): Even nowadays we give severe punishments when necessary at the time (Rivash 251)! It seems that Rav Yosef knew that Rav Chama did not intend for this, rather, to execute Torah law.

Chachmas Shlomo (52b): Really, Rav Chama did not err. It was a need of the time to burn her. He did not burn her with lead, like the Halachah, lest people say that Misas Beis Din applies nowadays. Rather, they will know that he burned her for a need of the time. I do not argue with the Gemara, for this makes no difference until Mashi'ach comes.

Rebuttal (Birkei Yosef 3): How can he say that the Gemara did not think of this?! Rather, they knew that Rav Chama intended to burn her according to law, or that people were not wantonly sinning and there was no need for a Hora'as Sha'ah (a temporary ruling). The Nimukei Yosef said that only the Great Sanhedrin can punish not according to Torah law. Rav Huna did not give the full Torah punishment. If the Maharshal argues with the Nimukei Yosef, he should have cited him and rejected him! Gidulei Terumah says that Rav Chama intended to burn her for Hora'as Sha'ah. His only mistake was that only the Great Sanhedrin can do so. Shevus Yakov says that the Nimukei Yosef means that only the Great Sanhedrin can give the full punishment of Misas Beis Din. It seems that Rav Chama intended for a greater punishment than the Torah (to burn her body). To resove the Nimukei Yosef, we can say that a Great Sanhedrin can punish greater than Torah law only when the person is not Chayav Misah, like the person who rode an animal on Shabbos. They killed him to scare people. Any Misas Beis Din scares people. There is no need to destroy the body. Since Rav Chama did so, we must say that he did not intend for Hora'as Sha'ah.

Birkei Yosef (3 DH Achen): We must say that Rav Chama erred about Rav Masnah's teaching. Even if he intended for Hora'as Sha'ah, if he knew that Mitzvah Sereifah does not destroy the body, he would not have destroyed the body for Hora'as Sha'ah. The Nimukei Yosef asked only what forced Rav Yosef to say that he erred also about Misah when there is no Kohen. Even though we could say that rather, he erred about who may punish unlike Torah law (he did not know that only a Great Sanhedrin can), in any case we must say that he erred about two matters. We learn from the Nimukei Yosef that nowadays a Beis Din cannot kill. Maharash Chivan says that nowadays Beis Din cannot do more than cut off a limb. Maharam Lublin (138) says that even if we explain the Nimukei Yosef simply, we do not follow him against all the Poskim. Further, perhaps the Nimukei Yosef means that only the Sanhedrin can give the Misah that the Torah gives, but any Beis Din can cause one to die in other ways. R. Yonah in Sanhedrin says that Rav Chama erred also about a third matter: only judges with Semichah can kill.

Rambam (Hilchos Sanhedrin 24:4): Beis Din can lash or kill someone who is not liable to receive lashes or death, not to transgress Torah law, rather, to make a fence for Torah. If Beis Din sees that people wantonly transgress a matter, they should fence and fix it as they see fit. This is a Hora'as Sha'ah. The Halachah is not fixed like this for all generations. Once they lashed a man who had Bi'ah with his wife under a fig tree.

Rambam (5): Similarly, Beis Din in any place and any time can lash someone of ill repute if everyone says that he transgresses Arayos. This is if the reports do not cease and he does not have known enemies that might be starting the reports. We disgrace a person of ill repute and revile his mother in front of him.

Kesef Mishneh: Rashi explains that he is lashed for "Al Beni Ki Lo Tovah ha'Shemu'ah", which is a Lav. This is an Asmachta.

Poskim

Shulchan Aruch (CM 2:1): Any Beis Din, even if it is did not get Semichah in Eretz Yisrael, if it sees that people wantonly transgress a matter (and it is necessary at the time) they can impose death or any other punishment, even if there is not full testimony. If the person is powerful, they may lash him through Nochrim. All their actions must be l'Shem Shomayim. Only the Gadol ha'Dor may do so, or Tuvei ha'Ir (the governors of the city) if Beis Din authorized them.

Beis Yosef (DH v'Gam): Once, it became known that a widow became pregnant from a Nochri. Judges decided to cut off her nose to mar her face, which she had adorned to entice adulterers. The Rosh consented.

Bedek ha'Bayis (DH Kosav, citing Zohar brought in Rikanti Reish Mishpatim): A judge who is meticulous to acquit Resha'im will die before his proper time. Beis Din should punish more than the Torah says to make a fence for Torah! A man once kissed an Eshes Ish, and Aba killed them.

Bach (2 DH u'Mah she'Chosav Rabeinu): The Rambam said 'similarly, Beis Din in any place and any time' to teach that even nowadays they can kill, as long as there is supporting evidence and unceasing rumors.

Birkei Yosef (3 DH veha'Zeh): The Tur understood that the Rambam holds that nowadays, Beis Din can kill. In Halachah 4, the Rambam said that Beis Din can lash or kill, and in Halachah 5, he said that Beis Din in any place and any time can lash. This implies that nowadays Beis Din cannot kill! Sha'arei Yeshu'ah says that the Tur had a different text of the Rambam. However, this text has needless repetition. I defend the Tur according to our text of the Rambam. The Rambam adopted the text of the Gemara; R. Eliezer ben Yakov said that Beis Din can lash and punish. The Rambam explained that this is in any place and any time to teach about lashes for ill repute. The Gemara did not say who can administer this. Even though if follows from a Kal va'Chomer from the ability to kill, the Rambam generally brings all the laws mentioned in the Gemara. Beis Din can kill only if people wantonly transgress, but they lash for ill repute even if only one person transgresses. 'Lash and punish' includes death. If the Rambam meant to exclude it, he would have explicitly said so!

Rema: This is the custom everywhere, that Tuvei ha'Ir in their city are like Beis Din ha'Gadol. They can lash, punish and make Hefker according to the custom. Some argue and say that Tuvei ha'Ir can only force the Tzibur to fulfill existing customs or what the entire Tzibur agreed to. In any case we follow the custom in the city, and all the more so if the city accepted the Tuvei ha'Ir to do everything.