Gemara

(Rabah bar bar Chanah): The first Tana teaches that if a man ate Chelev because he desired it, he is a Mumar;

R. (Yosi b'Rebbi) Yehudah says, even one who wears Sha'atnez is a Mumar.

They argue about Sha'atnez mid'Rabanan. Chachamim say that one is Mumar only for wearing Sha'atnez mid'Oraisa. R. Yehudah says, since this is a well-known Isur, he is a Mumar even for Sha'atnez mid'Rabanan.

Eruvin 69a (Beraisa - R. Yehudah): R. Gamliel said 'do your needs in the Mavoy (alleyway) before it gets dark and he (the Tzeduki who lives there, who rejects Divrei Chachamim) will forbid you (to carry in it on Shabbos).'

Inference: A Tzeduki is like a Nochri. (Bitul, i.e. renouncing ownership of his house, does not permit others to carry in the Mavuy.)

Contradiction (Mishnah - R. Yehudah): He told them 'do your needs in the Mavoy (before Shabbos), before (he will) Yotzi (take out) and forbid you.'

Answer: The Mishnah discusses a Tzeduki who is Mechalel Shabbos in private, the Beraisa discusses one who is Mechalel in public.

(Rav Huna): A Mumar is one who is Mechalel Shabbos in public.

69b (Rav Nachman bar Yitzchak): Rav Huna discusses who can give Reshus (Me'arev) and be Mevatel Reshus:

(Rav Ashi): Rav Huna holds like the following Tana, that Chilul Shabbos is as severe as idolatry.

(Beraisa): We accept Korbanos from sinners of Yisrael, to encourage them to repent, except for a Mumar, one who is Menasech (pours libations to idolatry), and one who is Mechalel Shabbos in public.

Rishonim

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.

Rosh (Eruvin 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 Perushim (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 is 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 says that after he takes out he cannot be Mevatel because he transgressed taking out without an Eruv, i.e. an Isur mid'Rabanan.

Poskim

Shulchan Aruch (OC 385:3): A Yisrael Mumar to idolatry or to be Mechalel Shabbos in public, even if he transgresses only mid'Rabanan Isurim of Shabbos, he is like a Nochri.

Beis Yosef (381 DH ha'Mevatel): Tosfos (69a DH Kan), the Rosh, the Mordechai and Rashi hold that one who transgresses mid'Rabanan Isurim of Shabbos cannot be Mevatel Reshus. Tosfos asked why Tzedukim can be Mevatel Reshus, and answered that 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. Even if he holds that one who does so is a Meshumad, he would not say so about Tzedukim nowadays, for they are not Mezid. They are like Ohnes; 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 (1): Many Poskim say that even Tzedukim nowadays who are Mechalel Shabbos in public are like Nochrim. Kara'im nowadays are Mechalel Shabbos in public at least regarding mid'Rabanan laws, so they cannot be Mevatel Reshus.

Aruch ha'Shulchan (EH 141:74): One who is Mechalel Shabbos in public, even regarding Isurim mid'Rabanan, is like a Meshumad, and he cannot be a Shali'ach to give a Get. The Magen Avraham (189:1) says that a Meshumad is one who seeks to anger Hash-m. If so, we can be lenient about one who does not intend to anger Hash-m. This requires investigation.

Shulchan Aruch (YD 2:5): A Mumar to be Mechalel Shabbos in public is like a Nochri (what he slaughters is a Neveilah).

Beis Hillel (3): Seemingly, for Shechitah he is like a Nochri only if he transgressed Shabbos mid'Oraisa. However, Tosfos in Eruvin says that he is called Mechalel Shabbos in public even for Isurim mid'Rabanan done repeatedly in front of 10 Yisre'elim. The Shulchan Aruch (OC 385:3) rules like this. The same applies to Shechitah. This is not only for Eruvin. The Halachah follows Rav Ashi, that he is like an idolater in every way.

Rebuttal (Pri Chodosh, Kuntres Acharon): There is no proof from there. Rav Ashi discusses one who does Melachos mid'Oraisa! Surely, one is not like a Nochri for Isurim mid'Rabanan, even of Shabbos. He is like a Nochri only regarding Eruv. One who is suspected of Isurim mid'Rabanan is not suspected of Isurim mid'Oraisa (Siman 119 below). Just like Rav Nachman bar Yitzchak holds that one suspected about Chilul Shabbos is not Meshumad for the entire Torah, just he cannot be Mevatel his Reshus. Rav Ashi agrees if he transgressed only Isurim mid'Rabanan.

Pischei Teshuvah (8, citing Mishnas Chachamim): One who transgressed Isurim mid'Rabanan of Shabbos or idolatry is not a Mumar to the entire Torah until he transgressed something with a Chiyuv Misah.

Shulchan Aruch (119:5): One suspected about a severe Isur is suspected about a light Isur, unless people are more careful about the light Isur.

Shulchan Aruch (7): One known to transgress an Aveirah other than idolatry or Chilul Shabbos in public, or does not believe in Divrei Chachamim, is believed about other Isurim.

Chazon Ish (YD 2:15): We hold like the first Tana of R. Yosi b'Rebbi Yehudah, that one does not become a Mumar for Kilayim mid'Rabanan. Tosfos says that one becomes a Mumar for Chilul Shabbos mid'Rabanan only regarding Bitul Reshus.

Har Tzvi (YD 7 DH Ulam): If one sends his son to a Nochri school in which they transgress Shabbos, is the father considered a Mechalel Shabbos? The Chasam Sofer (OC 83) says that the father does not transgress "do not do Melachah, you and your son...", for the son does not do for his father. Therefore, the father is not a Mechalel Shabbos, like the Pri Chodosh said. For Isurim mid'Rabanan of Shabbos he is like a Nochri only mid'Rabanan regarding Bitul Reshus. Tevu'os Shor (2:28) agrees. Beis Hillel says that even for mid'Rabanan Isurim of Shabbos he is called Mechalel Shabbos. Even he could admit that this does not apply to Aveiros of Chinuch, even if they pertain to Shabbos. Regarding the father it is not an Aveirah of Shabbos. Chinuch is a general Mitzvah not particular to Shabbos. The Ramban and Ran say similarly, that Lifnei Iver (even when it pertains to idolatry) is not an Aveirah of idolatry, so one need not forfeit his life for it. R. Akiva Eiger (on Shabbos Mishnah 1:4) says that one is a Mumar to the entire Torah even for Aveiros of Shabbos for which one is exempt but it is Asur mid'Rabanan, but not when it is Mutar regarding Shabbos and Asur due to Lifnei Iver. According to Mishnas Chachamim (cited above in Pischei Teshuvah), even if the father transgressed "do not do Melachah, you and your son...", he would not be a Mumar, for he is not Chayav Misah for this.

Igros Moshe (YD 2:5): Chachamim argue about whether one who is Mechalel Isurim mid'Rabanan of Shabbos in public is a Mumar. It is a Safek how we rule. If we can say that one who uses a microphone thinks that this is a Safek about a mid'Rabanan law, so he may be lenient (even though perhaps it is an Isur Torah), he is not Mezid, and he is not a Mumar. He probably knows that all Gedolim forbid. However, perhaps he thinks that he may rely on the lenient opinion, so this is like Chilul Shabbos in private. If one believes that Hash-m created the world in six days and commanded about Shabbos, just he cannot overcome his desire to sin, this is like one who transgresses due to lust, who is not a total Mumar. When he does so in public, he publicizes that he does not believe in creation. This does not apply if we know that he observes other laws of Shabbos.

Yabi'a Omer (1 YD 11): Most Acharonim hold that one who is Mechalel Isurim mid'Rabanan of Shabbos is not a Mumar.