Gemara

(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."

Hash-m (to Yehoshua): I may not tell the Halachos to you.

Bava Metzi'a 59a (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 said that if the law is like him, the carob tree, irrigation ditch, walls of the Beis Medrash, and Heaven should show it. Chachamim said that we don't bring proof from these matters.

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 or white hair came first, Hash-m said that he is Tahor (Musgar). The rest of the Heavenly academy said Tamei (Muchlat). They sent the angel of death to take Rabah to decide the matter. He died amidst saying 'Tahor, Tahor.'

Rishonim

Rambam (Hilchos Yesodei ha'Torah 9:1): "Lo va'Shamayim Hi" 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, he is a false prophet.

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 (4): If a Navi says that Hash-m said the Halachah follows Ploni, he is a false Navi, for he came to deny Torah - Lo ba'Shamayim Hi.

Or Some'ach: The Gemara says that Chachamim ruled like Beis Hillel against Beis Shamai because they were pleasant to each other and taught their colleagues' (Beis Shamai's) words (first). The Halachah follows Rav in Isur because he was frequently amidst Hora'ah (Rosh Bava Kama 4:4). We follow Shmuel in monetary matters, because he was frequent in the Reish Galusa's house and close to the judges (Bava Basra 65a). A Bas Kol said that the Halachah follows Beis Hillel and that the Halachah follows R. Eliezer everywhere. The Bas Kol was about the person, and not about a particular Halachah or oven. A Divine sign that a person is worthy to always rule like him does not show that he is correct about a particular law. Rather, it is like a general rule, that we should rule like Rav against Shmuel in Isurim. R. Yehoshua disagreed and said that we do not heed a Bas Kol. To decide a particular Halachah like Ploni, it says Lo ba'Shamayim Hi, like Yehoshua answered Yisrael. If a Navi said that the Halachah follows Ploni, he is a false Navi. The Bas Kol taught that it is proper to rule like Beis Hillel in general, just like it may say that it is proper to rule like a Chacham who is frequent in Hora'ah.

Rif (Sukah 15b): A Niktam (cut off) Hadas is Kosher even if a date did not grow.

Rambam (Hilchos Lulav 8:5): If the top of a Hadas was cut it is Kosher.

Rebuttal (Ra'avad): It is Pasul. Hash-m withheld the proper understanding of this from earlier Chachamim and gave me Ru'ach ha'Kodesh to understand it.

Ra'avad (Hilchos Lulav, (cited in Migdal Oz Hilchos Lulav 8:5) DH Inyan): A Mishnah of clearcut Halachah disqualifies Niktam Rosho. The entire Sugya connotes that it is Pasul. The Mishnah (34b) does not discuss Niktam Rosho, rather, Ketumim, i.e. upper branches cut off (from the bottom) while they are soft.

Defense (of Rif - Ramban's Hasagos on the Ra'avad): All the Ge'onim ruled like the Rif. Hadasim are "Anaf Etz Avos", which connotes branches cut off the tree!

Rebuttal (Ran, ibid.): Surey, the Sugya of Dichuy (33a, when a Hadas was Niktam and a date grew) is according to Halachah!

Magid Mishnah: The Ramban refuted the Ra'avad's proofs. Nevertheless, it is good to be concerned for the Ra'avad's opinion.

Ran (Drashah 7 DH v'Achshav, cited in introduction to Ketzos ha'Choshen DH u'Kevar): Why was there an argument between Hash-m and the Heavenly academy? Based on what they learned in their lives, it seemed that the law is Tamei, even though this is the opposite of "Emes" (Hash-m's opinion), just like those who argued with R. Eliezer did not retract due to Divine signs that the Halachah follows R. Eliezer. They sent for Rabah to show that human intellect can see that it is Tahor, just they had not attained this.

Ketzos ha'Choshen: They excommunicated R. Eliezer because it is improper to turn to Shamayim to resolve our disagreements. It is better to leave the matter in Safek. Hash-m refused to resolve Yehoshua's Sefekos.

Maharam Mintz (100, cited in Pischei Teshuvah CM 23:5): After a judge writes a ruling, he is not believed to explain it. We explain according to the interpretation of the words. If a judge were believed to explain his words, the Yeshiva above would surely have agreed with Hash-m, who wrote the Torah!

Rebuttal (Pischei Teshuvah, citing Maharha Yitzchaki): In Bava Metzi'a, the Chachamim of the generation had already decided, therefore Lo ba'Shamayim Hi.

Poskim

Shulchan Aruch (OC 46:6): Some have the custom to bless "ha'Nosen la'Ya'ef Ko'ach". This is improper.

Gra (DH v'Ein): We do not add new Berachos after the Gemara, like the Rosh says in Kidushin (1:41).

Rema: The custom spread throughout Ashkenaz to say it.

Magen Avraham (12): Keneses ha'Gedolah said that the Beis Yosef retracted at the end of his life and said to say it. Kavanos ha'Ari Zal says to say it.

Gra (DH Ach): The custom overrides Halachah in many places in Shas.

Sha'arei Teshuvah (6): Matzas Shimurim says that since the Rema agreed to say it, surely this was said to him through Ru'ach ha'Kodesh. Based on Kabalah, one must say it after Malbish Arumim.

Birkei Yosef (11): The custom has spread to say this, based on the Ari Zal. Even though we (Sefardim) accepted the Mechaber's rulings, had the Mechaber seen the Ari Zal's opinion to bless it, he would have agreed. Keneses ha'Gedolah says that the Mechaber retracted. Perhaps this was after the Ari Zal died.

Hagahah 15: Mar'is ha'Ayen writes that here the Birkei Yosef says that we follow the Ari Zal against the Mechaber. Pesach ha'Dvir (1:25:5) says that the Birkei Yosef ruled like the Ari Zal only because other Poskim rule like him and the Beis Yosef retracted. We have a tradition that all the secrets that the Ari Zal revealed are what he learned with Eliyahu ha'Navi. The Birkei Yosef said so in Chayim She'al 2:10:3. We need not publicize (deep secrets) to the masses, and also many Chachamim stick to Peshat, like the Birkei Yosef wrote in Kikar la'Aden 5:7. One who wants to do like the Ari Zal may, even though he does not understand the reason and secret of the matter, like the Birkei Yosef wrote above (25:11). For such matters, we do not say Lo ba'Shamayim Hi.

Shulchan Aruch (OC 646:1,10): If the top of a Hadas was cut it is Kosher, even if a date did not grow there. Some disqualify.

Rema: It is good to be stringent when possible.

Ha'aros on Birkei Yosef (2): The Ben Lev and R. Chaim Vital say that this was real Ru'ach ha'Kodesh, unlike Kapos Temarim 32b. L'Halachah it does not matter, for Lo ba'Shamayim Hi. The Ra'avad relied on Ru'ach ha'Kodesh because Ge'onim argued about this and we had no way to decide. If Shamayim revealed the ruling through Ru'ach ha'Kodesh to a Gadol ha'Dor, we do not say Lo ba'Shamayim Hi. The Ramban was not concerned for this, for he found a source in the Gemara to rule oppositely.

Birkei Yosef (654:2): Some women in Eretz Yisrael bless on the Lulav. This is wrong. They started this by themselves. We accepted the rulings of the Mechaber and Rambam; they say not to bless. Later, I saw that R. Yakov of Marvish (a Rishon cited in the Mordechai) asked about this from Shamayim, that it is a Berachah l'Vatalah. He was told 'did the generations improve?! Listen to everything Sarah says. Tell them "return to your tents and bless Hash-m." Megilah and Chanukah prove this.' Women bless on them because they were involved in the miracle. Likewise, Lulav shows that Yisrael have only one heart for Hash-m, and Shofar brings our commemoration in front of Hash-m. Also women need this! Therefore, they may bless on these.

Ha'arah (2): In Yosef Ometz (82), the Birkei Yosef explains that Lo ba'Shamayim Hi does not apply because we cannot decide between the Rishonim. We do not follow our understanding, rather, we compromise. It is better to follow the ones with whom Shamayim agreed. This is not making a new creed (which the Rambam forbids in Halachah 2), rather, solidifying the creed. Shem ha'Gedolim (entry R. Yakov of Marvish) says that when Chachamim of the generation cannot decide, they may ask Shamayim. Being stringent (due to Safek) is not Hora'ah (a ruling - Beitzah 2b). Indeed, for some matters our custom does not rely on Teshuvos Min ha'Shamayim, e.g. to bless on Hallel on Rosh Chodesh and Shehecheyanu on Purim. Rather, the Birkei Yosef teaches why we accept the custom of some women to bless. We do not say that they err. We follow Teshuvos from Shamayim only when there is a custom that perhaps preceded the Shulchan Aruch, e.g. to bless ha'Nosen la'Ya'ef Ko'ach, and the forms of the letters in a Sefer Torah, Tefilin and Mezuzos, R. Tam's opinion about (until when one may do Melachah on) Erev Shabbos, in which the Mechaber himself wrote not to change (from the custom). Had the Mechaber seen (what was shown from Shamayim), he would have agreed. Perhaps even Sefardi women had an old custom to bless on these Mitzvos.

Rav Pe'alim (2 OC 12 DH v'Da): I rely on the Ari Zal against the Shulchan Aruch (to permit Birkos Kri'as Shma after four hours). We find that the Shulchan Aruch holds that ha'Nosen la'Ya'ef Ko'ach is a Berachah l'Vatalah, and all of Yisrael bless it, based on the Ari Zal. Also, based on the Ari Zal one blesses Birkos ha'Shachar even if they do not apply to him (e.g. he did not hear a rooster, he is not wearing a belt), unlike the Shulchan Aruch.

Note: I heard in the name of ha'Gaon R. Yakov Hillel Shlita that the Ari Zal merited to learn from Eliyahu's Guf (his physical body), therefore 'Lo ba'Shamayim Hi' does not apply. The others who learned from Eliyahu saw him as an angel, so for them Lo ba'Shamayim Hi applies.