Gemara

Bar Kamtza reasoned that since Chachamim saw that he was disgraced and they did not protest, they approved. He decided to inform on them to the king.

Bar Kamtza (to the Keisar): The Jews have rebelled against you! Send a Korban, and see if they will offer it.

He sent a choice calf with Bar Kamtza. Bar Kamtza nicked it in a way that is not a blemish in a Nochri's Korban (on a private altar), but it is a blemish for Yisrael (or for offering on behalf of Nochrim in the Mikdash).

Chachamim thought to kill Bar Kamtza, lest he tell the Keisar that Yisrael did not offer his Korban. R. Zecharyah objected, lest people say that one who blemishes a Korban is killed.

Bava Kama 117a: Yosef wanted to show Yehudah's straw (to extortionists). Rav told him not to. He persisted 'I will show'. Rav Kahana broke Yosef's neckbone.

Rav: "...Like a trapped wild ox" - once a wild ox is trapped, no one has mercy on it. Similarly, once money of Yisrael falls into the hands of Nochrim, they have no mercy on it.

Berachos 58a: Rav Shila lashed a man (Ploni) who had relations with a Mitzris. The man informed on him to the king, that there is a Yisrael who judges without permission of the king. The king sent to take Rav Shila.

Rav Shila said that Ploni had had relations with a donkey. Eliyahu came, looking like a commoner, and testified to this. The court gave to Rav Shila a staff and authority to judge. He explained to Ploni that Mitzriyim are equated to donkeys -- "Asher Besar Chamorim Besaram."

Rav Shila saw that Ploni was going to report this to the king. Rav Shila reasoned: he seeks to kill me. The Torah says, if one seeks to kill you, kill him first! He killed him with the staff.

Shevuos 46a (Rav Nachman): If Reuven took an axe and said 'I will cut Shimon's tree', and we later see it cut and lying on the ground, we do not say that Reuven (surely) cut it and must pay. People often make empty, exaggerated threats.

Rishonim

Rambam (Hilchos Chovel u'Mazik 8:10): One may kill a Moser anywhere, even nowadays that we do not judge capital cases. One may kill him even before he informs. Once he says 'I will be Moser Ploni or his money', even a small sum, he permits himself to be killed. We warn him not to be Moser. If he is brazen and insists that he will, it is a Mitzvah to kill him. Whoever does so first, he merited.

Chazon Ish (YD 2:16): Why do capital cases need 23 judges, and all the special laws? Anyone who could be executed (received warning and still wanted to sin, so he) is a Mumar, who may be killed in any case! Perhaps we cast into a pit only when we cannot judge capital cases, but when we can, we do not cast until judging. One who ate Neveilah to anger Hash-m is lashed, and then we do not cast him into a pit. Further, we do not cast into a pit one who repented, but Teshuvah does not help for one sentenced by Beis Din. It seems that we do not cast until rebuking; perhaps he will accept rebuke. Perhaps we do not cast unless the person regularly does so. The Rambam says 'after it is publicly known...' It seems that a Beis Din of three must rule that one may cast a person into a pit. However, if Levi cast Ploni and afterwards Beis Din ruled that Ploni deserved this, Levi is exempt. However, the Rambam (Hilchos Mamrim 3:2) does not require Beis Din, only that it is publicly known. It seems that we cast only when Hash-m's Hashgachah is evident in the world, and there are open miracles and voices from Heaven. Then all knew that sins bring afflictions to the world. However, when people are weak in Emunah, casting people in pits seems sinister in their eyes. The entire purpose is to fix things! When it will not, we rather show love to try to get sinners to repent.

Note: The question is not so clear to me. R. Chananel (cited in Rivash 238) says that we do not (overtly) kill one who was Moser, but we cast him into a pit. The Gemara explains 'we do not lift such people from a pit', i.e. we find excuses so it will not seem like we intend to kill him. Presumably, we likewise cast into a pit as if it were an accident! Also, when we can give Misas Beis Din, it is a Mitzvah to do so! I have not seen other Poskim say that the law does not apply nowadays (that we lack open miracles and voices from Heaven and people's Emunah is weak). Some texts in Avodah Zarah say 'Hayu' (they used to cast into pits). It seems that this is due to censorship, just like many texts say 'Hefker Beis Din Hayah Hefker.'

Question: How can we kill a Moser? It is a Safek if he will really do so. Perhaps he merely makes an exaggerated threat, but will not do so, like Shevuos 46a!

Answer #1 (Yam Shel Shlomo Bava Kama 10:50): Tosfos (46a DH Avid) cites the Roke'ach to say that when he is established to do so, we assume that he really will. (Tosfos discussed cutting trees. The Yam Shel Shlomo understands that the same applies to Moser.)

Answer #2 (Rashba 1:181): A Moser is like a snake. Whoever kills him first, he merited. If he regularly does so, once he says that he will, it is as if he already did. If one seeks to kill you, kill him first, like Rav Shila. He did not wait to see if Ploni will really do so! Even if he does not regularly do so, if we recognize that he intends to do so, it is as if we know that he will do so. He is killed due to this Safek. Even others may save the victim by killing the Moser, just like a Rodef.

Poskim

Shulchan Aruch (CM 388:10): One may kill a Moser anywhere, even nowadays. One may kill him even before he informs. Once he says 'I will be Moser Ploni or his money', even a small sum, he permits himself to be killed. We warn him not to be Moser. If he is brazen and insists that he will, it is a Mitzvah to kill him. Whoever does so first, he merited.

SMA (28): The Rivash (236) says that if one was seized for being Moser, or saying things for which is he is Chayav Misah, we do not accept Arevim to assure that he will not flee. If he fled, what would we do to the Arevim?

SMA (29): Even though a Moser does not overtly damage money, once money of Yisrael falls into Nochri hands, they have no mercy on it. They seek pretexts to the point that there is often mortal danger. Therefore, a Moser is like a Rodef.

Yam Shel Shlomo (ibid.): When a Yisrael's money falls into their hands, they take some today, and ultimately all of it. They torture him, hoping that he will admit that he has more money. A Moser is the ultimate Rodef!