Gemara

31a (Beraisa - R. Yehudah): One may be Me'arev with Terumah in a grave.

Chachamim forbid acquiring Shevisah with Isurei Hana'ah (a grave).

R. Yehudah holds that Mitzvos Lav Leihanos Nitnu. (Fulfilling Mitzvos is not considered benefit regarding Isurei Hana'ah.)

Question: Rava taught that Mitzvos Lav Leihanos Nitnu. Do the Tana'im argue about this?!

Answer: No, all agree that Lav Leihanos Nitnu. R. Yehudah holds that we are Me'arev only for a Mitzvah. Chachamim permit even for Reshus (an optional matter).

Question Rav Yosef taught that we are Me'arev only for a Mitzvah. Do Tana'im argue about this?

Answer (for Rav Yosef): No. All agree that we are Me'arev only for a Mitzvah, and that Mitzvos Lav Leihanos Nitnu. They argue about whether he wants the food to be guarded even after he acquired Shevisah [and benefits from the grave].

36a (Mishnah): One may place one Eruv in the east, and one in the west and stipulate... If Chachamim come from both directions, I will go where I want.

R. Yehudah says, if one of them is his Rebbi, he may go only in that direction.

Chachamim hold that sometimes a person prefers his friend over his Rebbi.

82a (Mishnah): To make Eruv Techumim for many, one designates a barrel and says 'this is for everyone in the city who wants to go to the mourner's house or to the wedding feast.'

(Rav Yosef): One may be Me'arev only in order to [go to] do a Mitzvah.

Question: What is the Chidush? The Mishnah says '... to the mourner's house or to the wedding feast'!

Answer: One might have thought that the Mishnah discusses a normal case, but it applies also to other cases. Rav Yosef teaches that this is not so.

Pesachim 49a (Mishnah): If one was on his way to a Mitzvah, i.e. to slaughter Korban Pesach... if he has time to return, eradicate his Chametz, and return to the Mitzvah, he must do so;

If not, he does Bitul in his heart [and does the Mitzvah].

If he was going to make Shevisah of Reshus, he returns immediately.

Rishonim

Rambam (Hilchos Eruvin 6:6): We make Eruv Techumim only for a Mitzvah, e.g. to go to a mourner's house... or to greet his Rebbi or friend who arrived, and similar matters. Or, it is permitted due to fear, e.g. to flee from Nochrim or bandits or similar matters. If one was Me'arev for Reshus, it is valid.

Terumas ha'Deshen (77): The Ge'onim permit Eruv Techumim only for a Mitzvah. If one wants to go for Tiyul (an outing) on Yom Tov in his orchard, which will give him Simchah, this is called a Devar Mitzvah. The Rosh and Hagahos Maimoniyos bring from R. Tam that we take a minor out [to Reshus ha'Rabim] if his father wants to go for a Tiyul for Simchas Yom Tov, and he cannot leave the child alone in the house. This is considered Hotza'ah for the needs of Yom Tov, just like Hotza'ah of a baby to circumcise him or a Sefer Torah to read in it. It seems that everything is considered a Devar Mitzvah to make an Eruv for it, for the Ramban permits making an Eruv in order to greet his Rebbi or friend coming from the road. The Rambam permits going through water on Yom Kipur only to greet one's Rebbi, father, or someone greater in Chachmah than himself. He did not mention a friend. The Rosh says similarly in the name of the Ritz Gei'us. One cannot bring a proof from there. One may pass through water even to guard Peros, which is not a Mitzvah at all! In any case, greeting a friend does not count for anything, but one may be Me'arev for this. This shows that we are lenient about Eruv.

Rashi (49a DH Lishbos): The Mishnah discusses one going to the end of the Techum at the end of Erev Yom Tov, to acquire Shevisah there, to be able to go to a matter of Reshus. If he was going to acquire Shevisah for a Mitzvah, this is like going to slaughter Korban Pesach.

Tosfos (49a DH Lishbos): According to Rashi, our Mishnah is difficult for the opinion that permits Eruv Techumim only for a Mitzvah! Rather, he was going to rejoice on Pesach in the house of his friend or relative.

Rashi (82a DH Ein Me'arvin): Chachamim permitted to leave the Techum through an Eruv only for a Devar Mitzvah.

Hagahos Ashri (Eruvin 8:1, citing Or Zaru'a): Seemingly, the Halachah follows Rava, who permits to be Me'arev for Reshus. According to Rav Yosef, if one was Me'arev for Reshus, even b'Di'eved it is invalid! If one made an Eruv for a Mitzvah, Rashi connotes that he may not rely on it for Reshus. Perhaps it is permitted. Also the Ramak permits. Or Zaru'a brings from the Rambam that it is valid.

Poskim

Shulchan Aruch (OC 415:1): We make Eruv Techumim only for a Mitzvah, e.g. to go to a mourner's house... or to greet his Rebbi or friend who arrived, and similar matters.

Beis Yosef (DH u'Mah she'Chasav Oh): The Rambam permits Eruv Techumim even in order to greet his friend. It seems that this is only a friend who is a Chacham, and he will learn from him. The Mishnah about stipulating about an Eruv connotes like this. However, the Terumas ha'Deshen connotes that he permits even if the friend is not a Chacham. He permits even for Tiyul for Simchas Yom Tov. This is considered a Mitzvah.

Gra (DH Oh Chavero): This is like it says in Eruvin 36b, that sometimes a person prefers his friend.

Mishnah Berurah (3): Even if he is not a Chacham, it is considered a Mitzvah because he came from the road.

Kaf ha'Chayim (6): Eliyahu Rabah and Aruch ha'Shulchan say that the Rashba and Shiltei ha'Giborim hold like the Beis Yosef, that it is only if his friend is a Chacham, and this is primary. Others bring the Magen Avraham, who is lenient.

Magen Avraham (2): It is permitted also if he went to oversee communal matters (Tosfos 82a DH Ein), or if he was outside his city, and needed to go to his house (Shiltei ha'Giborim 27a:1). Maharal was unsure whether one may be Me'arev go to a Minyan. Based on what I wrote in 90:6, surely it is permitted. (Some say that one must walk a Mil to pray with a Minyan, or four Mil in the direction he was walking. This shows that it is a great Mitzvah - Machatzis ha'Shekel.)

Mishnah Berurah (4): If one was outside his city, and wants to go to his house to pray on Shabbos or Yom Tov, this is a Mitzvah. One may go to redeem captives, help deliver a baby or save from Nochrim even without an Eruv.

Kaf ha'Chayim (8): The Magid Mishneh brings that the Ramban holds that going to his house is not a Mitzvah, but the Acharonim rule like Shiltei ha'Giborim, who says that it is.

Magen Avraham (2): One who is Me'arev with his legs may be Me'arev for Reshus (409:7).

Mishnah Berurah (1): Some permit to be Me'arev with his legs l'Chatchilah even for Reshus. Some do not distinguish an Eruv through bread or through his legs. It is permitted l'Chatchilah only for a Mitzvah. When needed, one may rely on the first opinion.

Kaf ha'Chayim (1): R. Yehonason, the Magen Avraham and Chayei Adam permit. Tosfos, the Rosh and Ran do not distinguish, and the Yerushalmi connotes like this. This is why the Shulchan Aruch omitted this. One may be lenient only for a great need.

Rema: One who wants to go for Tiyul on Yom Tov or Shabbos in his orchard, which gives him Simchah, is called a Devar Mitzvah.

Gra (DH Oh she'Rotzeh): We learn from Beitzah 12a, like Tosfos (DH Hachi) explains. (One may take a minor in Reshus ha'Rabim on Yom Tov for Tiyul. We permit even a ball for Tiyul!)

Shulchan Aruch (ibid.): Or, it is permitted due to fear, e.g. to flee from Nochrim or bandits.

Mishnah Berurah (6): This is if the Nochrim will pain him or steal money. If there is danger, he may flee without an Eruv.

Rema: Then it is permitted even for Reshus.

Beis Yosef (DH u'Mah she'Chasav u'Mihu): The Tur says that once one made an Eruv for a Mitzvah [he may rely on it even for Reshus]. Semak connotes like this. He cites Perek b'Chol Me'arvin. I did not find a source there. Rashi (82a) connotes unlike this. Hagahos Ashri is astounding. We do not find any Amora who argues with Rav Yosef! On 31a, we answer for Rava that really, the Tana'im argue about an Eruv for Reshus, and Chachamim permit. The Halachah follows Chachamim, and we follow Rava against Rav Yosef, for he is Basra. I disagree. The Gemara showed how it is possible for all Tana'im to agree with Rava's law. We do not rely on this to reject Rav Yosef's teaching, which was taught where this Halachah is discussed, and no one argued there. The Gemara questioned what is Rav Yosef's Chidush, for it is clear from the Mishnah!

Mishnah Berurah (7): Whenever one made an Eruv for a Mitzvah, he may go 2000 Amos from it even for Reshus.

Kaf ha'Chayim (11): The Shulchan Aruch did not need to teach this, for he says that even if one was Me'arev for Reshus, it is valid.

Shulchan Aruch (ibid.): If one was Me'arev for Reshus, it is valid.

Beis Yosef (DH Kasav): The Magid Mishneh says that the Ramban agrees with the Rambam, that if one was Me'arev for Reshus, it is valid. Rashi and the Rashba hold like Bahag [that it is invalid]. The Rashba says that this is only when he was Me'arev with bread. If he was Me'arev with his legs, all agree that he acquired Shevisah even for Reshus. I say that one may do so l'Chatchilah. The Isur to be Me'arev for Reshus was said only regarding bread, which should not work based on letter of the law. It is a mere leniency for rich people (49b). Chachamim were lenient only for the sake of a Mitzvah.

Gra (DH v'Im): This is like Rashi (49a). See Tosfos (49a).

Damesek Eliezer: Rashi permits only b'Di'eved. According to the Beis Yosef, Rashi could agree about an Eruv with bread, for Rashi explains that he was Me'arev with his legs!

Note: It is difficult to say that the Mishnah discusses someone heading to the Techum [when there is still time for Bitul, i.e. before the sixth hour] to stay there until nightfall! Surely, Rashi explained so because he holds that one may be Me'arev for Reshus only with his legs!