Gemara

Runya was Ravina's planter. He caused a loss, and Ravina fired him. Runya complained to Rava, who defended Ravina.

Runya: He did not warn me (that he would fire me if I cause a loss)!

Rava: He need not warn you!

(Rava): The following may be fired (for negligence) without warning: one who teaches children, a planter, a Tovach (slaughterer), a bloodletter, and the town scribe.

The general rule is, in any trade in which a mistake cannot be corrected, it is as if the worker was warned.

Bava Basra 21a (Rava): If there are two teachers, and one learns more but the other teaches more precisely, we prefer the former. If he makes any mistakes, the children will later learn the correct version.

(Rav Dimi of Neharde'a): No, we take the latter. Once a child learned incorrectly, the mistake will stick with him.

Rishonim

Rif (Bava Basra 10b): A teacher she'Posha (was negligent) with the children is like one who was warned.

Nimukei Yosef (Bava Metzia 66a DH Peseida): The teacher taught incorrectly, hit the children too much, or neglected to teach them.

Rambam (Hilchos Sechirus 10:7): The following may be fired without warning: one who plants for the city and caused a loss...

Ra'avad: The same applies to one who plants for an individual.

Magid Mishneh: The Rambam learns from Runya, who was Ravina's planter. Rava ruled that Ravina was allowed to fire him without warning. Perhaps the Rambam holds that Runya planted for the Rabim, including Ravina. The last profession that Rava listed was 'Sofer Masya.' The word 'Masya' (of the city)' applies to all the professions listed.

Bach (CM 306:12): All the others listed are people whom the Tzibur appointed over them, so presumably a planter is the same. The Gemara says that he was Ravina's planter because he was the head of the city, and they did not do anything without his consent. Ravina hired Runya.for the entire city. We require a Chazakah to fire someone, so we must say that he caused a loss to two or three people, according to Rebbi and R. Shimon ben Gamliel (who argue about whether two or three times are needed to make a Chazakah).

Rambam (ibid): ...A city Tovach who did invalid Shechitah, a bloodletter who wounded...

Rashba (citing the Ra'avad, brought in Magid Mishneh): If the Tovach is paid, what loss results from his mistake? He must pay for the loss! If he works for free, he does not lose if he is fired! We must say that even if he works for free, we fine him, and he may not slaughter for free or for pay. Alternatively, he was paid, and caused a great loss. Even though he pays for the animal, he does not pay for the owner's embarrassment (the host has no meat to serve to his guests) or the guests' embarrassment. Also, sometimes much meat is needed, and he is the only Tovach. Even though they need not be warned, they need Chazakah (to cause a loss). We do not remove them until there is a Chazakah or they are warned.

Note: Another answer is suggested from Rivash (113). If there is a Safek whether or not a Tovach slaughtered properly, the meat is forbidden but we cannot force him to pay. The Lechem Mishneh questioned his reasoning, for he understood that the the animal was Safek Tereifah even before he slaughtered it, so surely the Tovach is exempt!

Ha'Emek She'elah (92:1): The Ra'avad holds that we discuss one who works for an individual, therefore, if he caused a loss once, he can say that he erred or was Shogeg. He is fired only after he was established to err. If not, no one can keep a job! The She'altos, Bahag, Rif, Rambam hold that we discuss only one who works for the city. 'Musrin v'Omdim' means that he is warned at every moment; he cannot say that he erred. Perhaps the Rambam and Ra'avad agree about the Halachah follows. They argue only about what Rava discusses.

SMA (CM 306:20): The answer for a Tovach does not apply to a planter! Perhaps we can say that he harmed the tree he planted, and had he planted properly, it would have already produced Peros. We cannot obligate him for these Peros, for we do not know how much it would have produced.

Rambam (ibid): ... a scribe who erred in documents, and a teacher she'Posha with the children and did not teach or taught incorrectly.

Magid Mishneh: The Rif explains that the teacher neglected to teach the children. Rashi says that he taught to them incorrectly. The Rambam brings both Perushim. Meforshim explain that Rashi discusses a teacher who himself errs, but if he knows, just he did not supervise a child who said wrong, we do not fire him.

Tosfos (109b DH v'Sofer): Rashi's Perush is wrong, for Rava (Bava Basra 21a) says that the children will later learn the correct version.

Note: Rashi must hold that Rav Dimi's opinion was taught last because the Halachah follows him. Presumably, Tosfos rules like Rava because he was greater, or because he himself taught the prfoessions that can be fired without warning.

Ohel Moshe (Bava Basra 21b): Tosfos agrees that we fire a teacher if he himself errs. He argues with Rashi about a teacher who knows, but does not supervise to see that the child erred, like Yo'av's teacher (Bava Basra 21a).

Rambam (ibid): The same applies to any profession in which the loss cannot be returned. It is as if they were warned to strive to work (properly), since the Tzibur appointed them.

Rosh (9:38): Runya complained (only) about being fired, for he received the improvements he made to the field.

Beis Yosef (CM 306 DH u'Mah she'Chosav Rabeinu): The Tur learned from here that one pays the improvements that the worker made.

Hagahos Ashri: A children's teacher is considered a mistake that cannot be corrected, for while he was teaching incorrectly, he could have taught something else. This implies that all the more so, if he was idle at the time (when he should have been teaching) for a day or two, the teacher can be fired without warning. Rebbi once did so.

Tosfos (Bava Basra 21b DH Sofer): Rashi explains that a Sefer Torah written incorrectly cannot be corrected. This is wrong! Rather, Sofer refers to one who writes documents.

Poskim

Shulchan Aruch (CM 306:8): The following may be fired without warning: one who plants for the city and caused a loss...

Rema: Some say that the same applies to one who plants for an individual.

SMA (19): They argue about all of the professions.

Shulchan Aruch (ibid): ... A city slaughterer who did invalid Shechitah, a bloodletter who wounded...

Gra (21): This is like the Rambam, Tur and Rashi in Bava Basra. Rashi in Bava Metzia says that 'Uman' refers to a Mohel. Both of these Perushim are true.

Shulchan Aruch (ibid): ... A scribe who erred in documents, and a teacher of children who was negligent and did not teach (Rema - even one or two days) or taught incorrectly.

Beis Yosef (DH Shasal): Teaching incorrectly is irreversible, for once the mistaken entered the Talmid's head, it will stay.

SMA (21): It suffices to say that he did not teach for one day! Perhaps he adopts the text of Shemos 21:21 to allude to a 24-hour period.

Shulchan Aruch (ibid): The same applies to any profession in which the loss cannot be returned. It is as if they were warned to strive to work (properly), since the Tzibur appointed them.

Rema: Some say that even though they need not be warned, they need Chazakah. We do not remove them until there is a Chazakah or they are warned.

Beis Yosef (DH v'Al): The Nimukei Yosef (66a DH v'Tavcha) says that we do not remove them until they are Muchzakim three times, or are warned.