The halachah is once a Jew, always a Jew. The source most often cited for this is ישראל אף על פי שחטא ישראל הוא (Sanhedrin 44a)--A Jew, even if he has sinned remains a Jew. Rashi in a teshuvah cites this Gemara as the basis for saying that a mumar retains his Jewish status. Thus a Jew cannot lose his obligation to perform mitzvot. The fact that he may lose his status as a Jew in good standing and thus is ineligible to write a sefer Torah or perform shechita is irrelevant to his obligation in mitzvot.
Rabbi Joseph B. Soloveitchik explained this by reference to the two types of kedushah that a Jew possesses, symbolized by the two blessings, שלא עשני גוי and אשר בחר בנו מכל העמים. One holiness, that of שלא עשני גוי, is a product of being born a Jew (or converting to Judaism). It cannot be renounced. However, there is a second kedushah which comes from observing the Torah, אשר בחר בנו מכל העמים ונתן לנו את תורתו; regarding this element of kedushah, if a Jew does not observe the mitzvot he loses his status (Chumash Mesoras Harav Vayikra, p. 223):
One aspect of our chosenness relates to being the progeny of Abraham,
and it is for this aspect which we recite the blessing who has not
made me a gentile. A second chosenness comes through the study of
Torah, reflected in the blessing who has chosen us from among the
nations. This chosenness involves a higher sanctification than is
afforded through simple lineage. By understanding the duality of our
kedushah, we can explain a puzzling area of Jewish law. The halachic
status of a Jew who renounces his religion is complex: although his
status as a Jew is compromised, it does not entirely disappear. With
regard to specific halachos, an apostate Jew is considered no
different from a non-Jew. For example, regarding the slaughter of
animals or the writing of Torah scrolls, an apostate is considered to
be a non-Jew: his meat would be considered non-kosher, and his Torah
scroll would be invalid. On the other hand, if he betroths a woman,
his betrothal would be valid. How can we understand the hybrid nature
of the halachos concerning an apostate? The apostate Jew retains the
sanctity afforded through his lineage—a heritage that cannot be
renounced, no matter what he does. This kedushah applies equally to
the apostate as it does to the greatest of rabbis. The second
kedushah, however, is dependent on the sanctity of the individual
attainable only through limud haTorah. The apostate has renounced this
second kedushah.
