Can a Jew lose their status as being a Jew by performing any action (example: performing idolatry, disavowing core beliefs, converting to another religion, etc)? Or once a person is a Jew then they keep this status no matter what?
And if a person would always maintain their status as a Jew (in terms of other Jews viewing that person to still be obligated to follow the commandments), then what in practice would be the significance of a Jew converting to another religion?
For a person who has renounced their Judaism, it appears that while the person doesn't need to undergo a full conversion (on a Torah level), there is a custom to reaccept the Torah on some level and immerse in a mikvah before returning to the previous status. This seems to be especially true if the Jew literally worshiped idolatry. However, they are still a Jew in every sense of the word. (See https://aish.com/who-is-a-jew/) and (https://aish.com/status-of-baptized-jew/)
If the person converted to another religion, there may also be procedural differences depending on what religion the person converted to (since some religions may have the status of being idolatrous, and some may not), or what they specifically did to disown their Judaism (or what beliefs they held that were against Jewish core beliefs).
Of course, this is a complicated topic with additional responsa literature, so please see a proper rabbinic authority if this is a practical issue for you.
Further reading:
While I am not qualified to answer the question directly, this is not a new question. For some related sources, see the subject of Anusim on Wikipedia, specifically, and especially, the short section on Rabbinic Legal Opinions, here:
"Hakham Se‘adyá ben Maimón ibn Danan, one of the most respected Sephardic Sages after the Expulsion, in the 15th century stated:
Indeed, when it comes to lineage, all the people of Israel are brethren. We are all the sons of one father, the rebels (reshaim) and criminals, the heretics (meshumadim) and forced ones (anusim), and the proselytes (gerim) who are attached to the house of Jacob. All these are Israelites. Even if they left God or denied Him, or violated His Law, the yoke of that Law is still upon their shoulders and will never be removed from them.[4]
Hakham BenSión Uziel, the Chief Sephardic Rabbi of the State of Israel, stated in mid-20th c.
And we still have to clarify on the (subject of) Anusím, to whom the government forbids them to perform Halakhicly valid marriages, if it's necessary to say that their wives must have a Get to permit them (to marry another man), for the reason that, by force of the Law (Hazaqáh), a man does not have intercourse for promiscuity (zenút). . . (In our very case), we deal with those who converted and kept Torah in secrecy and hide their religion because of the gentile surveillance, we say that they do have intercourse for the sake of marriage.
It follows that Hakham Uziel considered anusím as Jews, because only Jews can give or receive a Get, a Jewish divorce.
Rabbi Moshe ben Maimon (Maimonides) stated in the Mishneh Torah Sefer Shofetím, Hilekhót Mumarím 3:3
But their children and grandchildren [of Jewish rebels], who, misguided by their parents . . . and trained in their views, are like children taken captive by the gentiles and raised in their laws and customs (weghidelúhu haGoyím `al dathám), whose status is that of an ’anús [one who abjures Jewish law under duress], who, although he later learns that he is a Jew, meets Jews, observes them practice their laws, is nevertheless to be regarded as an ’anús, since he was reared in the erroneous ways of his parents . . . Therefore efforts should be made to bring them back in repentance (LeFikakh rawí leHah zirán biTeshubáh), to draw them near by friendly relations, so that they may return to the strength-giving source, i.e., the Toráh "
Certain sins will cause a Jew to lose his status for many things, but not for marriage, divorce and children. I.e, a Jewish woman's children are always Jewish no matter what sins she's done, and they do not require conversion. However, if a person or group becomes completely disconnected from Judaism for many generations, they will completely lose their Jewish status for everything.
Rambam gives a list of 13 foundations of the religion. At the conclusion of this list he writes:
וכאשר יהיו קיימים לאדם כל היסודות הללו ואמונתו בהם אמתית הרי הוא נכנס בכלל ישראל וחובה לאהבו ולחמול עליו וכל מה שצוה ה' אותנו זה על זה מן האהבה והאחוה ואפילו עשה מה שיכול להיות מן העבירות מחמת תאותו והתגברות יצרו הרע הרי הוא נענש לפי גודל מריו ויש לו חלק והוא מפושעי ישראל וכאשר יפקפק אדם ביסוד מאלו היסודות הרי זה יצא מן הכלל וכפר בעיקר ונקרא מין ואפיקורות וקוצץ בנטיעות וחובה לשנותו ועליו הוא אומר הלא משנאיך ה' אשנא וכו (Qafih translation)
And when a person upholds all these foundations, and his belief in them is true, he enters into the group of Israel and it is an obligation to love him, and to have compassion for him, and all that Hashem commanded us regarding love and brotherhood for each other. And even if he has done what can be of the sins due to his desires and the overpowerment of his evil inclination, he will be punished commensurate to the greatness of his rebellion but he still has a portion and he is of the sinners of Israel. But when a person doubts one of these foundations he has left the group, and denied God, and is called a sectarian and a heretic and a cutter of shoots, and it is an obligation to hate him, and of him it is said "do I not hate those who hate you, Hashem?"
We found in the Talmud (Yev 17a), about descendants of the lost tribes that could become a great problem:
לא זזו משם עד שעשאום נכרים גמורים
"They did not move from there before they made them [=Sages considered them] full gentiles"
For a great article on this topic see Is a Jew Who Converts Still Jewish http://www.chabad.org/library/article_cdo/aid/1269075/jewish/Is-a-Jew-Who-Converts-Still-Jewish.htm A brief Quote: "Achan has broken “My covenant that I have commanded them”—interpreted by the Talmud to mean not only one detail, but the entire covenant of Torah. Yet he remains not only a Jew, but “Israel”—the entirety of the Jewish People in a single individual.
The principle extends not only to genealogical Jews, but converts as well. In Tractate Yevamot we learn that once a person has fulfilled all the requirements of a proper conversion, he is considered “like Israel in all matters.” The Talmud explains those last words to mean that even if this convert would return to his pagan ways, “if he marries a Jewish woman, he has the same status as an apostate Jew, and they are considered married.” Based on the above statement of the Talmud, the Jewish Code of Law rules that a marriage between a Jewish man and a Jewish woman who “convert out” is completely valid. Therefore, their children are considered Jewish and could also marry other Jews."
From my perspective, "Once a Jew Always a Jew;" because the Convert comes into the Community by vowing to uphold the 613 Mitzvoth (Commandments). Numbers 30:2 "When a man makes a vow to the LORD or takes an oath to obligate himself by a pledge, he must not break his word but must do everything he said."
If he fails to keep the Mitzvot in their entirety, he will be punished like any other Jew (except he return [repent or make tshuva] to Hashem) whereas if he keeps the mitzvot in their entirety, he will be rewarded like any other Jew.
A good source to read about converts is, Converts and Conversion to Judaism, by Rabbi Josef Lifland.
Today it is hard for a Jew to lose his status as a Jew since Rabenu Ovadia Yosef says that this generation has the status of tinok shenishbah.
On the other hand there are sins that have the punishment of caret, so his soul would be cut off from the Jewish people after his death.
Even though someone might maintain his status as Jew, he may gain the status of someone that can't be trusted (with food or testimony).
There is a concept of "cutting off from Israel". I.e Shemot 12:15. Whoever eats leaven on Passover is cut off from Israel. Such person is definitely not Jewish. There are other similar prohibitions in the Torah. I.e Shemot 30:33, 38, Vaikra 7:20. Etc.
