Gemara

Rebbi permits Eruv Techumim [in a tree] because he could take the food Bein ha'Shemashos, since Shevus (Isurim mid'Rabanan) do(es) not apply Bein ha'Shemashos.

(Mishnah): If an Eruv was in a tree below 10 Tefachim, it is valid.

This is like Rebbi.

34b (Mishnah): If he put the Eruv on top of a reed that was detached and later inserted into the ground, it is valid.

Question (Rav Ada bar Masnah): If it is still attached, it is invalid. This is like Chachamim, who forbid Shevus Bein ha'Shemashos.

Shabbos 34a (Mishnah): If it is Safek dark, one may not tithe Vadai (definite) Tevel or immerse Kelim. One may tithe Demai.

Rishonim

Rambam (Hilchos Eruvin 6:9): If one intended to be Shoves in Reshus ha'Yachid or Reshus ha'Rabim, and left an Eruv in Karmelis, or vice-versa, it is valid. An Eruv acquires Shevisah Bein ha'Shemashos, and then one may transfer from one of these Reshuyos to Karmelis for the sake of a Mitzvah. Chachamim did not decree any Isurim mid'Rabanan Bein ha'Shemashos regarding a Mitzvah or in pressed circumstances.

Rambam (Hilchos Shabbos 24:10): All matters forbidden due to Shevus, Chachamim did not decree about them Bein ha'Shemashos, if needed for a Mitzvah or in pressed circumstances. E.g. Bein ha'Shemashos one may alight a tree or swim to bring a Lulav or Shofar. One may take an Eruv down from a tree or from Karmelis. Similarly, if one was distracted and frantic Bein ha'Shemashos and needed something forbidden due to Shevus, it is permitted. If there was no pressure or Mitzvah, it is forbidden. Therefore, one may not tithe Vadai Tevel Bein ha'Shemashos, even though this is Shevus, but one may tithe Demai.

Magid Mishneh: A Stam Mishnah is like Rebbi, who permits Shevus Bein ha'Shemashos. Why does a Mishnah forbid tithing and immersing Kelim Bein ha'Shemashos? These are Shevus! They answered that Chachamim permitted Shevus only for a Mitzvah, such as Eruv, like the Rambam says. The Rambam added one who was distracted and frantic. I say that it is because he holds that if one was Me'arev for Reshus, it is valid. Even so, Chachamim said that they did not decree about Shevus regarding it. This shows that anyone in pressed circumstances Bein ha'Shemashos is permitted. Some ask from the Isur to make Eruv Techumim Bein ha'Shemashos. Eruvin 6:13 answers this. (It is invalid because perhaps it is already night.)

Rashi (Shabbos 8b DH v'Rebbi): Rebbi holds that Chachamim decreed Shevus only during Vadai Shabbos, but not Bein ha'Shemashos.

Rashba (Eruvin 32b DH Hachi): Just like Rebbi permits Shevus Bein ha'Shemashos of Erev Shabbos, he permits Bein ha'Shemashos of Motza'ei Shabbos.

Poskim

Shulchan Aruch (OC 342:1): Anything forbidden mid'Rabanan, Chachamim did not decree about it Bein ha'Shemashos if it is for the sake of a Mitzvah or in pressed circumstances.

Beis Yosef (DH Kol): Rebbi permits an Eruv in a tree, because Shevus does not apply Bein ha'Shemashos. This implies that this is only for a Mitzvah, for we permit making Eruv Techumim only for a Mitzvah (82a). If not for a Mitzvah, Chachamim decreed Bein ha'Shemashos. This is why one may not tithe Vadai Tevel or immerse Kelim Bein ha'Shemashos, even though these are Shevus. The Magid Mishneh says so. The Tur's intent is not clear. He permits 'if one was distracted and needed to tithe Peros Bein ha'Shemashos.' If this is Vadai Tevel, the Mishnah forbids it. If it is Demai, it is permitted even without distraction! Perhaps he explains that the Mishnah forbids Vadai when he does not need it for Shabbos. What is needed for Shabbos is a Mitzvah. Why did we permit Demai, i.e. not for Shabbos? We must say that we permit in pressed circumstances. The Tur permits only Demai. I prefer to say that even Vadai is permitted, even not for Shabbos, if he is distracted and frantic, since we permit Shevus Bein ha'Shemashos in pressed circumstances. The Mishnah that forbids Vadai is like the opinion that forbids Shevus Bein ha'Shemashos, like Rashi explained (Shabbos 34a DH Safek).

Mishnah Berurah (1): Really, some Isurei mid'Rabanan are forbidden Bein ha'Shemashos even for a Mitzvah, like in 409:3, or carrying less than four Amos at a time. Chachamim were more stringent, for one is prone to come to a Torah Isur. Many Poskim forbid Eruv Techumim Bein ha'Shemashos, even though it is for a Mitzvah. Melachah she'Einah Tzerichah l'Gufah is more stringent than other Isurei mid'Rabanan, and it is always forbidden Bein ha'Shemashos, and all the more so according to the Rambam, who forbids mid'Oraisa. This entire Sa'if does not discuss one who accepted Shabbos. If he accepted [through saying Mizmor Shir or Lecha Dodi], even if it is still day, or even if only the Tzibur accepted Shabbos, Shabbos takes effect on him against his will. He may not do any Shevus by himself, even for a Mitzvah, if not through a Nochri.

Kaf ha'Chayim (6): One may not tell a Nochri to extinguish a Ner Bein ha'Shemashos if not for a Mitzvah or one who is distracted and frantic.

Magen Avraham (1): It is not clear whether Chachamim decreed in Bein ha'Shemashos of Motza'ei Shabbos. The end of the day is different than the beginning of the day. The Kedushah does not leave due to Safek, like I wrote in 386 (12,13) and the beginning of 562.

Birkei Yosef (2): The Rashba equates Bein ha'Shemashos of Erev Shabbos and Motza'ei Shabbos.

Mishnah Berurah (2): The Chayei Adam is stringent on Motza'ei Shabbos. The Beis Meir says that letter of the law one may be lenient for the sake of a Mitzvah, but in practice one must be stringent because we are not experts about when is Bein ha'Shemashos.

Kaf ha'Chayim (8): The Birkei Yosef and most Poskim are lenient. Therefore, on Bein ha'Shemashos of Motza'ei Shabbos one may tell a Nochri to light a lamp for the sake of a Mitzvah, e.g. Seudah Shelishis or prayer.

Bi'ur Halachah (DH Mutar Lo): The Pri Megadim asked why we permit all of Bein ha'Shemashos. We should forbid some, for Tosefes Shabbos. One must add from Chol to Kodesh, according to the opinion that this is mid'Oraisa! Perhaps indeed, it is forbidden shortly before night due to Tosefes Shabbos.

Shulchan Aruch (ibid.): E.g. Bein ha'Shemashos one may alight a tree or swim to bring a Lulav or Shofar. One may take an Eruv down from a tree or from Karmelis.

Question (Taz 1): If the Mishnah is like the one who forbids Shevus Bein ha'Shemashos, why does the Tur permit only one who needed to tithe, i.e. for the needs of Shabbos? The Bach permits even for the need of Shabbos only if he is distracted, for this is not such a Mitzvah. He can manage without it (the Peros). This is unlike a Lulav. Semag permits for the need of a Mitzvah or in pressed circumstances, e.g. he was distracted and frantic. This connotes anything that will cause pain if he cannot do it. I explain the Tur like this. He permits Demai. He discusses distraction regarding other matters. Maharshal (Teshuvah 46) says that for a great need or [to avoid] a big loss, one may tell a Nochri Bein ha'Shemashos. Even on Shabbos we are lenient for a big loss to say 'whoever saves [will not lose].' Therefore, they did not decree about this Bein ha'Shemashos. He heard that a Chacham permitted telling a Nochri Bein ha'Shemashos to light a Yartzite lamp for one who forgot to light it during the day. I explain that the Rambam permits in pressed circumstances even for Reshus.

Mishnah Berurah (3): Even if he does not need the Eruv now, it is considered for a Mitzvah, for the Eruv is for a Mitzvah, and if he cannot take the Eruv, it is invalid.

Kaf ha'Chayim (7): We may permit vows Bein ha'Shemashos, but one may not annul vows, for perhaps it is already night.

Kaf ha'Chayim (10): Whatever is permitted for the sake of a Mitzvah is permitted for the sake of others (Rema 333:1).

Bi'ur Halachah (DH Mutar Bein): Something needed for Shabbos that he could manage without it, but it will increase Oneg Shabbos, is considered a Mitzvah.

Shulchan Aruch (ibid.): Similarly, if one was distracted and frantic Bein ha'Shemashos and needed something forbidden due to Shevus, it is permitted.

Mishnah Berurah (5): The Gra (261:1) explains that the Rambam permits for one who is distracted and frantic, i.e. greatly pressed circumstances, e.g. he wants to be Me'arev in order to flee from a Nochri. Then he permits an Eruv on a tree or in a Karmelis.

Sha'ar ha'Tziyun (5): The Gra agrees that it is permitted to avoid a big loss.

Shulchan Aruch (ibid.): This is why Bein ha'Shemashos one may tell a Nochri to light Ner Shabbos.

Mishnah Berurah (7): He asks the Nochri to light immediately. He may not ask him to light on Shabbos, even if he asked him during the day (307:2). The Mechaber permits not only for the need of a meal. He is lenient even to light in a room where he will sleep.