Can a Jew lose his status as Jew?

  Rambam (Avoda Zara 2:5)An Israelite who worshiped idolatry, behold him, he is in everything to be judged as an idolater, and not as an Israelite who violates a prohibitive commandment punishable by stoning. A convert to idolatry, behold him, he is rebellious against the whole scope of the Torah. Likewise are infidels of among Israel not to be judged in aught as Israelites, nor should they ever be received as penitents, for it is said: "None that go unto her return, neither do they attain unto the paths of life" (Prov. 2.19). The infidels are they that absorb themselves in the swerving fancies of their heart about the brutish matters hereinabove spoken of, as a consequence whereof they transgress by stepping upon the vitals of the Torah with spite, soul-sickening and high-handedness, and proclaim that therein lies no iniquity. It is forbidden to converse with them, and to answer any argument concerning them, as it is said: "And come not nigh the door of her house" (Ibid. 5.8). As a rule the thought of an infidel drifts toward idolatry. 

Rambam (Commentary to 10th chapter Sanhedrin) 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?" 

Mamrim (3:2-3)מֵאַחַר שֶׁנִּתְפַּרְסֵם שֶׁהוּא כּוֹפֵר בַּתּוֹרָה שֶׁבְּעַל פֶּה [מוֹרִידִין אוֹתוֹ] וְלֹא מַעֲלִין וַהֲרֵי הוּא כִּשְׁאָר כָּל הָאֶפִּיקוֹרוֹסִין וְהָאוֹמְרִין אֵין תּוֹרָה מִן הַשָּׁמַיִם וְהַמּוֹסְרִין וְהַמּוּמָרִין. שֶׁכָּל אֵלּוּ אֵינָם בִּכְלַל יִשְׂרָאֵל וְאֵין צָרִיךְ לֹא לְעֵדִים וְלֹא הַתְרָאָה וְלֹא דַּיָּנִים [אֶלָּא כָּל הַהוֹרֵג אֶחָד מֵהֶן עָשָׂה מִצְוָה גְּדוֹלָה וְהֵסִיר הַמִּכְשׁוֹל]: בַּמֶּה דְּבָרִים אֲמוּרִים בְּאִישׁ שֶׁכָּפַר בַּתּוֹרָה שֶׁבְּעַל פֶּה בְּמַחֲשַׁבְתּוֹ וּבִדְבָרִים שֶׁנִּרְאוּ לוֹ. וְהָלַךְ אַחַר דַּעְתּוֹ הַקַּלָּה וְאַחַר שְׁרִירוּת לִבּוֹ וְכוֹפֵר בַּתּוֹרָה שֶׁבְּעַל פֶּה תְּחִלָּה כְּצָדוֹק וּבַיְתּוֹס וְכֵן כָּל הַתּוֹעִים אַחֲרָיו. אֲבָל בְּנֵי הַתּוֹעִים הָאֵלֶּה וּבְנֵי בְּנֵיהֶם שֶׁהִדִּיחוּ אוֹתָם אֲבוֹתָם וְנוֹלְדוּ בֵּין הַקָּרָאִים וְגִדְּלוּ אוֹתָם עַל דַּעְתָּם. הֲרֵי הוּא כְּתִינוֹק שֶׁנִּשְׁבָּה בֵּינֵיהֶם וְגִדְּלוּהוּ וְאֵינוֹ זָרִיז לֶאֱחֹז בְּדַרְכֵי הַמִּצְוֹת שֶׁהֲרֵי הוּא כְּאָנוּס וְאַף עַל פִּי שֶׁשָּׁמַע אַחַר כָּךְ [שֶׁהוּא יְהוּדִי וְרָאָה הַיְהוּדִים וְדָתָם הֲרֵי הוּא כְּאָנוּס שֶׁהֲרֵי גִּדְּלוּהוּ עַל טָעוּתָם] כָּךְ אֵלּוּ שֶׁאָמַרְנוּ הָאוֹחֲזִים בְּדַרְכֵי אֲבוֹתָם הַקָּרָאִים שֶׁטָּעוּ. לְפִיכָךְ רָאוּי לְהַחְזִירָן בִּתְשׁוּבָה וּלְמָשְׁכָם בְּדִבְרֵי שָׁלוֹם עַד שֶׁיַּחְזְרוּ לְאֵיתָן הַתּוֹרָה: 

Yevamos (17a)

The Gemara asks: Aren’t there Jewish girls who were captured by gentiles, whose children are considered to be Jews? And Ravina said: Learn from this that the son of your daughter from a gentile is called your son. If so, the descendants of Jewish women captured by gentiles would indeed be Jews. The Gemara answers: This is no concern, as it is learned as a tradition that the girls from the ten tribes of that generation became barren and did not give birth to any offspring, whereas some of the exiled men of the ten tribes married gentile women. Consequently, all of the children born there were gentiles.

Yevamos (17a) The Gemara asks: Aren’t there Jewish girls who were captured by gentiles, whose children are considered to be Jews? And Ravina said: Learn from this that the son of your daughter from a gentile is called your son. If so, the descendants of Jewish women captured by gentiles would indeed be Jews. The Gemara answers: This is no concern, as it is learned as a tradition that the girls from the ten tribes of that generation became barren and did not give birth to any offspring, whereas some of the exiled men of the ten tribes married gentile women. Consequently, all of the children born there were gentiles. There are those who say that Rav Yehuda actually related the following: When I said this halakha before Shmuel, he said to me: They did not move from there, the place where they deliberated on this matter, until they rendered all of them, including those who intermingled with the ten tribes in different locations, full-fledged gentiles.