Gemara

31b (Mishnah): If one sent an Eruv with one who does not acknowledge [the enactment of] Eruvin, it is Pasul.

(Rav Chisda): This refers to Kusim (Nochrim whom Sancheriv settled in place of the 10 exiled Shevatim. They later converted, but did not follow Divrei Chachamim.)

61b (Mishnah): If a Yisrael lives in a Chatzer with a Nochri, or with one who does not acknowledge Eruv, the latter forbids the Yisrael [to transfer between his house and the Chatzer, unless he rents the Nochri's Reshus for Shabbos];

R. Gamliel says, a case occurred in which a Tzeduki lived with us in a Mavoy in Yerushalayim. (Tzedukim do not heed Divrei Chachamim.) My father told us 'quickly take Kelim into the Mavoy before he does and forbids you!'

R. Yehudah says, he told them 'do your needs in the Mavoy, before he will take out into the Mavoy and forbid you.'

68b: The Mishnah is abbreviated. It means as follows:

A Tzeduki is like a Nochri;

R. Gamliel says, a Tzeduki is not like a Nochri;

R. Gamliel says, a case occurred... My father told us 'quickly take Kelim into the Mavoy before he does and forbids you!'

69a (Beraisa - R. Yehudah): He said 'do your needs in the Mavoy before it gets dark and he will forbid you.'

Inference: A Tzeduki is like a Nochri. (Bitul does not help.)

Contradiction (Mishnah - R. Yehudah): He told them 'do your needs... before [he] will be Yotzi (take out) and forbid you.' (His Bitul helps until he takes out!)

Answer #1: He meant '...before the day will be Yotzei (end, and Shabbos will begin).

Answer #2: The Mishnah discusses a Tzeduki who is Mechalel Shabbos in private. The Beraisa discusses one who is Mechalel in public. (He is like a Nochri.)

A certain man went out [on Shabbos] with a band of fragrances. When he saw R. Yehudah Nesi'ah, he covered it.

R. Yehudah Nesi'ah: This person can be Mevatel according to R. Yehudah [since he is ashamed for me to see that he is Mechalel Shabbos].

(Beraisa): A Yisrael Mumar can be Mevatel Reshus only if he keeps Shabbos in public.

A Nochri cannot be Mevatel. He must rent his Reshus to permit carrying.

Rishonim

Rif (21a): One who is Mechalel Shabbos in the market is like a Nochri, so he cannot be Mevatel Reshus.

Rambam (Hilchos Eruvin 2:6): If a Yisrael is Mechalel Shabbos in public, he is like a Nochri in every way. He cannot be Mevatel his Reshus.

Rambam (16): If a Yisrael is Mechalel Shabbos in public, or serves idolatry, he is like a Nochri in every way. We cannot be Me'arev with him, and he cannot be Mevatel his Reshus. If he does not observe Shabbos in the market, he cannot be Mevatel his premises, only a Yisrael can give permission or be Mevatel his premises. Regarding a Nochri, one must rent the premises from him. If he does not serve idolatry, and is not Mechalel Shabbos, e.g. Tzedukim and those who deny oral Torah, if he does not acknowledge the Mitzvah of Eruv, we are not Me'arev with him, but one may not rent from him, for he is not like a Nochri. He can be Mevatel his Reshus to a Kosher Yisrael. This is how to fix [the Chatzer].

Rosh (6:13): Rabbeinu Meir says that a Tzeduki is like a Nochri, for he transgresses mid'Rabanan laws of Shabbos. This is like the first Tana. Even though R. Meir argues, the Halachah follows R. Yehudah against R. Meir. I disagree, In Nidah (33b) we say that wives of Tzedukim fear the Chachamim. Also the Ri says that Tzedukim fear Chachamim and do not transgress in public, so they are not considered Nochrim.

Rosh (14): Even if one transgresses Shabbos through Isurim mid'Rabanan, he is like a Nochri and he cannot be Mevatel his Reshus. We distinguished between Chilul Shabbos in private and in public, but not between an Isur mid'Oraisa and an Isur mid'Rabanan! Rashi (69a DH b'Chumreta) says that he cannot be Mevatel because he transgressed Hotza'ah (transferring from Reshus to Reshus) without an Eruv, i.e. an Isur mid'Rabanan.

Mordechai (516): The Halachah is that a Tzeduki is like a Nochri, for he transgresses mid'Rabanan matters in public. Even though R. Meir disagrees according to R. Gamliel, R. Yehudah holds that he is like a Nochri. Even according to the Ri, who says that R. Meir holds like R. Gamliel, the Halachah follows R. Yehudah against R. Meir. Even though the Halachah follows the lenient opinion in Eruvin, this is not considered a leniency, for there is also a stringency. If a Tzeduki is like a Nochri, he cannot be Mevatel Reshus, but one may rent from him. If he is not like a Nochri, he can be Mevatel Reshus, but one may not rent from him.

Poskim

Shulchan Aruch (OC 385:1): A Tzeduki is like a Yisrael, He can be Mevatel his Reshus.

Beis Yosef (DH Tzeduki): Maharam rules that a Tzeduki is like a Nochri. The Rosh and Rambam disagree.

Taz (1): In Sa'if 2, the Mechaber rules that one who is Mechalel Shabbos in public, even mid'Rabanan matters, is a Mumar. In any case, Tzedukim nowadays are not considered Mumarim, even though they transgress mid'Rabanan matters in public. Their Mezid is like Ones, for they merely continue their fathers' actions. This is like a Tinok she'Nishba (a baby captured by Nochrim. He does not know that he transgresses.)

Gra (DH Tzeduki): This is like R. Gamliel, and even according to R. Yehudah, in the second answer.

Damesek Eliezer: The second answer is the conclusion. We discuss a Tzeduki who is not Mechalel Shabbos in public, for if he were, he cannot be Mevatel (Sa'if 3).

Mishnah Berurah (1): Tzedukim lean towards Apikorsus. They do not believe in oral Torah. Even so, they are like Yisre'elim regarding Bitul, even though they are Mechalel Shabbos in public, at least for mid'Rabanan laws. Even though such a person is like a Nochri (Sa'if 3), Tzedukim are different. They merely persist in their fathers' ways (Beis Yosef). Several Poskim hold that even a Tzeduki who is Mechalel Shabbos in public is like a Nochri. This is relevant to Kara'ites nowadays. Since they are Mechalel Shabbos in public at least for mid'Rabanan laws, they cannot be Mevatel. One must rent from them.

Kaf ha'Chayim (2): R. Betzalel Ashkenazi, the Mabit, Shach, Pri Chodosh, Korban Nesan'el and Shiltei ha'Giborim citing Ri'az, say that nowadays Tzedukim are like Nochrim [even though they merely persist in their fathers' ways]. The only solution is to rent from them.

Shulchan Aruch (ibid.): However, his Eruv does not help, since he does not acknowledge Eruv.

Gra (DH Aval): The reason he cannot be a Shali'ach for an Eruv (31b) is because he cannot be Me'arev, like it says in Gitin (22b, that one who is not in the Parshah of Gitin cannot be a Shali'ach to receive a Get).

Mishnah Berurah (2): Even if he gave towards the Eruv, it does not help. He forbids until he is Mevatel his Reshus to them. This applies even to one who believes in oral Torah, if he does not acknowledge Eruv.

Shulchan Aruch (3): If a Yisrael is a Mumar to idolatry, or to be Mechalel Shabbos in public, even for only an Isur mid'Rabanan, he is like a Nochri. If he is Mechalel only in private, even regarding an Isur mid'Oraisa, he is like a Yisrael, and he can be Mevatel Reshus.

Beis Yosef (DH u'Mah she'Chasav Afilu): Tosfos (69a DH Kan) explained that Tzedukim can be Mevatel Reshus because they keep even mid'Rabanan laws of Shabbos, for they fear Chachamim. Nowadays, the Tzedukim transgress mid'Rabanan laws of Shabbos, so they cannot be Mevatel. The Rambam brought the Beraisa verbatim, and did not mention Chilul Shabbos mid'Rabanan. He holds that Tzedukim are not Mezid. They are like Ones; they merely follow their fathers' ways. This is like a baby captured by Nochrim, like he wrote in Perush ha'Mishnayos (Chulin 1:1).

Mishnah Berurah (6): If one is ashamed to be Mechalel Shabbos in front of a great person, even if he is Mechalel in front of many people, this is considered in private.

Bi'ur Halachah (DH Oh): The Shulchan Aruch discusses one who transgresses because he casts off the yoke of Shamayim. If he errs and thinks that a matter is permitted, presumably, he is not a Mumar.