Gemara

(Beraisa): R. Eliezer and Chachamim argued about Tum'ah of a kind of oven. R. Eliezer gave all proofs in the world; Chachamim did not accept them.

R. Eliezer: If the law is like me, this carob tree, the irrigation ditch, the walls of the Beis Medrash, and Heaven should show it. (The tree moved 100 Amos, the water reversed the direction of its flow, and the walls leaned to fall.)

Chachamim said 'we don't bring proof from trees or irrigation ditches.' R. Yehoshua rebuked the walls for interfering in a dispute of Chachamim.

A Bas Kol (voice from Heaven) said 'do not argue with R. Eliezer. The Halachah always follows him!' R. Yehoshua said "Lo va'Shamayim Hi."

(R. Yirmeyah): Once the Torah was given on Sinai, we do not heed a Bas Kol. Rather, we follow the majority of Chachamim - "Acharei Rabim Lehatos".

86a: If we are unsure whether the Tzara'as (white appearance on the skin) or white hair came first, Hash-m said that he is Tahor (Musgar). The rest of the Heavenly academy said Tamei (Muchlat). The angel of death was sent to take Rabah to decide the matter. He died amidst saying 'Tahor, Tahor.'

Megilah 2b (R. Yirmeyah): Nevi'im taught the five final-letters of the Aleph-Beis.

Question: "Eleh ha'Mitzvos" - a Navi may not change a law through prophecy (changing the letters affects Tefilin and Mezuzos)!

Answer: All the letters were already known, just it was not known which are the final letters and which come in the beginning or middle of words.

Question: Still, this is something new, so a Navi may not teach it!

Answer: It was known and forgotten. The Nevi'im just restored it.

Yevamos 14a: Reish Lakish says that in practice, Beis Shamai did not follow their own teachings. R. Yochanan says that they did.

Question: Before the Bas Kol (announced that the Halachah follows Beis Hillel), all should agree they followed their teachings! After the Bas Kol, all should agree they did not follow their teachings!

Answer #1: They argue about before the Bas Kol. Reish Lakish says that Beis Shamai conducted like Beis Hillel, for Beis Hillel were the majority. R. Yochanan says that they followed their own teachings, for they were sharper.

Answer #2: It was after the Bas Kol. Reish Lakish says that Beis Shamai heeded the Bas Kol. R. Yochanan holds like R. Yehoshua, who does not heed a Bas Kol.

Shabbos 108a - Question (Mar brei d'Ravina): May one write Tefilin on skin of a Tahor fish?

Answer (Rav Nachman bar Yitzchak): If Eliyahu will come and say.

Question: What does this mean? (A Navi cannot resolve Halachah!)

Answer: If he will say that (after it is tanned) it is no longer considered disgusting (like it was before tanning), it is Kosher; if not, not.

Zevachim 62a (Rabah bar bar Chanah): Three prophets came back with the exiles from Bavel. One testified that the Mizbe'ach may be up to 60 Amos, one testified where it should be, and one testified that we may bring Korbanos (if the Mizbe'ach is in its proper place) even without a Beis ha'Mikdash.

Temurah 16a (Rav Yehudah): In the mourning over Moshe's death, 3000 Halachos were forgotten. Yehoshua said that he cannot ask Hash-m, for "Lo va'Shamayim Hi." Shmuel said that he cannot ask Hash-m, for "Eleh ha'Mitzvos" - a Navi may not teach a new (mid'Oraisa) law through prophecy.

Rishonim

Rambam (Hilchos Yesodei ha'Torah 9:1): It is clear and explicit in the Torah that it is an eternal Mitzvah. There can be no change, detraction or addition. It says "everything that I command you, you will oberve. Do not add to or detract from it" and "eternal statutes for your generations." It says "Lo va'Shamayim Hi." This teaches that a Navi cannot make new laws. Therefore, if a Yisrael or Nochri will do a miracle and say that Hash-m sent him to add or remove a Mitzvah, or give an explanation of a Mitzvah that we did not hear from Moshe, or that the Mitzvos Yisrael were commanded were temporary, he is a false prophet, for he contradicts Moshe's prophecy. We kill him for deliberately speaking in Hash-m's name what he was not commanded. Hash-m commanded Moshe that this Mitzvah is forever. Hash-m is not a person, who lies!

Kesef Mishneh: R. Yirmeyah explained that we do not heed a Bas Kol to decide Halachah. All the more so, we do not heed a Navi's prophecy!

Rambam (Hilchos Tzara'as 2:9): If we are unsure whether the Tzara'as or white hair came first, he is Tamei. It seems that it is a Safek.

Kesef Mishneh: We do not rely on what Hash-m said, for Lo va'Shamayim Hi. Rabah died while saying 'Tahor', so Lo va'Shamayim Hi applies and we cannot learn from him.

Tosfos (59b DH Lo): In Yevamos we say that the Halachah follows Beis Hillel due to the Bas Kol. Here we do not heed a Bas Kol, because it opposed Divrei Torah - "Acharei Rabim Lehatos". There, the Bas Kol supported Divrei Torah, for Beis Hillel were the majority. The only doubt was because Beis Shamai were sharper. Also, here the Bas Kol was only for the honor of R. Eliezer, who requested a proof from Shamayim. There, we said that R. Yochanan holds like R. Yehoshua, who does not heed a Bas Kol. Our only source is here (where there are reasons not to heed it)! Since R. Yehoshua said "Lo va'Shamayim Hi", this shows that he holds that we never heed a Bas Kol.

Maharsha (DH Lo): R. Yehoshua must hold that the Halachah follows Beis Hillel because they were the majority.

Maharsha (Temurah 16a DH Omar): Yehoshua had already learned the laws, so he did not say "Eleh ha'Mitzvos", for they were not new for him.

Rashi (Shabbos 108a DH Mai): We cannot learn from Eliyahu what is permitted or forbidden, for Lo va'Shamayim Hi.

Poskim

Birkei Yosef (OC 32:4 DH v'Nir'eh): A Navi cannot tell us the Halachah through prophecy, but he can say things from his Chachmah. When the Gemara says 'if Eliyahu will come and say... we will not heed him', it means if he will say through prophecy. We may not resolve a Safek law through Eliyahu. The Rambam (introduction to Perush ha'Mishnayos) says similarly that if a Navi testifies that Hash-m told him a law about a certain Mitzvah or that a certain reasoning is true, he is a false prophet and we kill him. Hagahos R. Betzalel (Gitin 6) asks, since Eliyahu frequently spoke with Amora'im, why didn't he resolve Teiku (unsettled questions left for Eliyahu)? He answered that the Eliyahu who frequented the Amora'im was Pinchus, and in the future a different Eliyahu will resolve Teiku. The Rashba (Menachos 45a) asked why Eliyahu will need to explain difficult verses in Yechezkeil. When Eliyahu comes, the dead will be revived, and Yechezkeil himself can answer! He answered that perhaps Eliyahu will come before Techiyas ha'Mesim. Or, before Mashi'ach Eliyahu will be able to explain it in a Beis Medrash; he was regularly in Rebbi's Beis Medrash. How can R. Betzalel and the Rashba say that Eliyahu will explain laws? We can also ask against Hagahos Maimoniyos, Sefer ha'Chinuch and Beis Yosef, who rule like R. Shimon, that Nochrim have no Tum'as Ohel, because Eliyahu said so to Rabah bar Avuha (Bava Metzia 114b). They must hold that Eliyahu said so based on his awesome Chachmah, not through prophecy. Rashi in Shabbos means that we would not heed what Eliyahu says through prophecy. However, what was the Gemara's question against Rav Nachman bar Yitzchak? Perhaps he meant, if Eliyahu will come and say through his Chachmah we will heed him!

Mishneh l'Melech (Hilchos Ishus 9:6): We may rely on Eliyahu to resolve a Safek in Metzi'us (which of two men is a Kohen - Tosfos Yevamos 99b), but not a Safek in Halachah. However, perhaps this is only for money, but not for Safek Isur.

Birkei Yosef (ibid.): Tosefes Yom ha'Kipurim (Yoma 75a) says that we cannot rely on prophecy to say which father a baby was from, or that Ploni sold a slave, for the Torah required witnesses. He asked why the Gemara says that the manna resolved such doubts (the true father or master would get extra manna for him), and from Shabbos 34a (R. Shimon was Metaher certain places through Ru'ach ha'Kodesh). He answered that Moshe used to rule according to Din Torah, and the manna would verify his ruling. R. Shimon relied on his tradition. Tosefes Yom ha'Kipurim holds that prophecy cannot resolve even a Safek in Metzi'us. Prophets who returned from Bavel testified about the size and location of the Mizbe'ach, and that we may bring Korbanos even without a Mikdash (Zevachim 62a). Even though these are Metzi'us, the Nevi'im needed to testify. Had they said them through prophecy, we would not accept it. It seems to me that we may rely on a Navi for a Safek in Metzi'us, e.g. whether fish skin ceases to be repulsive. We may rely on Eliyahu to teach about Kevi'us (Berachos 35b), whether Terumah became Tamei (Pesachim 15), who gave the bigger deposit (Bava Metzia 37a), the final letters (Megilah 3a), whether a Yevamah will miscarry (Yevamos 35b) whether an animal's innards are lean (Tosfos Menachos 64a), and which day was made Rosh Chodesh in Eretz Yisrael (Tosfos Beitzah 30b and Rosh Sof Sukah). The Rashba (Teshuvah 10) says that David was allowed to take Michal back, for he knew through prophecy that Michal never had Bi'ah with Palti. All these show that we may rely on Nevi'im for a Safek in Metzi'us. In Zevachim the Nevi'im knew testimony, so it was proper to testify! The Halachos forgotten during the mourning over Moshe were as if they were never taught, so if a Navi would teach them, it would be like a new law (Temurah 16a).