What is the Jewish approach to including non-observant Jewish people in the ritual or practice of the Jewish religion?
This is a very intriguing and somewhat intricate question, which I’ll only be able to address in general terms within this venue.
In a nutshell, the basic spirit of Judaism is to include non-observant Jews in the community, and in ritual and practice. However, certain mitzvot require observance by the observant, while certain very severe transgressions exclude one from the community and communal observance.
Regarding the basic inclusive spirit of Judaism, several teachings illustrate this point:
One of the mitzvot of the holiday of Succot involves binding together and waving a branch of a date-palm, three myrtle branches, two willow branches and a citron fruit. The Talmudic Sages note (Lev. Raba 30:12) that regarding taste and scent, which correspond to Torah and mitzvot respectively, these items have either both, one or neither qualities.
Thus, the citron, which has both taste and scent, corresponds to Jews who are learned and observant; the date-palm, whose fruit has taste but no scent, corresponds to those who are learned but lack observance; the myrtle, which has scent but no taste, corresponds to those who are observant but lack Torah knowledge; the willow, which has neither taste nor scent, corresponds to those who have neither. However, just as the willow is nevertheless bound with and included in this mitzvah, so too Jews void of Torah and mitzvot are to be included with everyone else.
Similarly, at the onset of the holiest day of the year, the Day of Atonement, on which the final verdict for life or death, health or sickness, prosperity or poverty is decreed upon every individual, a special service, the moving Kol Nidrei prayer, publicly announces the community’s willingness to include the transgressors in the communal prayer.
A third such source is regarding the special incense whose particular ingredients could only be prepared and offered in the Holy Temple on the Golden Altar before G‑d. This incense was renowned for its beautiful scent, as befitting to honor G‑d, and is described by the Torah as being “sweet-smelling before the L-rd”. In fact, of its eleven ingredients, ten were very aromatic, while one, chelbana (gelbanum), had an unpleasant odor. The Talmudic Sages taught that this demonstrates that including the unobservant pleases G‑d (Karetut 6a, see Rashi Ex. 30:34).
That being said, certain mitzvot require participation exclusively of observant individuals.
For example, even though prayer may be recited personally and individually, certain aspects or forms of prayer require ten adult males and the reciting of specific, fixed liturgy. While an unobservant, Jewishly unknowledgeable man could constitute one of the ten for parts of prayer that require only the presence a minyan (for example, kaddish), he could not do so for prayers which require communal recitation (for example, amida or kedusha) if he does not recite what’s needed along with everyone else.
Another example involves creating an enclosed, “private” space which permits carrying on Shabbat within an otherwise public domain in which it is forbidden to carry. What unites otherwise disparate homes, families and individuals into this carrying space is that they join as one “family” of Sabbath observers where this space becomes their shared “home”. However, Jews living in that area who do not recognize the observance of Shabbat prevent the unification of Sabbath observers, and thus hinder the formation of this area, defined by the Eruv.
Additionally, certain very severe prohibitions separate those who transgress them from the community and disqualify their participation in communal ritual and practice. For example, wanton, public Sabbath desecration or idol worship prevent one from being considered a part of the community and would disqualify the person’s ritual slaughter or writing of ritual texts such as a Torah scroll, tefillin or mezuzot until he properly and publicly repents these sins.