A person who does not observe Yom Tob, but observes Shabbat, can be counted toward a Minyan. One who violates Yom Kippur, however, is akin to a Shabbat violator. As discussed in a previous installment, the status of a public Shabbat violator vis-à-vis being counted for a Minyan is subject to considerable debate and discussion.
A member of the Karaite sect cannot be counted for a Minyan. This sect acknowledges the written text of the Torah as divine law but rejects the authority of the Torah She'be'al Peh – the oral tradition interpreting the Torah. Thus, for example, the Karaites wear Tefillin between their eyes, because they follow the literal meaning of the requirement to wear Tefillin "Ben Enecha" ("between your eyes"), rejecting the Sages' understanding that this refers to the spot on the top of the head parallel to the area between the eyes. Likewise, they follow the literal meaning of the Torah's command not to kindle a flame on Shabbat (Shemot 35:3), and so they keep the lights and heating off on Shabbat, sitting in the dark and cold and eating cold food. We, who accept the Rabbis' interpretation of this verse, as forbidding only the act of kindling on Shabbat, but not having a preexisting flame, specifically make a point of lighting candles before Shabbat to illuminate the home, and of eating hot dishes on Shabbat. Since Karaites heretically deny the authority of our oral tradition, they are considered gentiles with respect to a Minyan, and cannot be counted.
Needless to say, if a Karaite or any other kind of heretic or sinner performs Teshuba, they are treated as full-fledged observant Jews and certainly count for a Minyan.
An observant Jew who does not read Hebrew, and thus cannot pray, may be counted toward a Minyan, as long as he understands the concept of Tefilla. If, for example, a Jew who did not receive a religious education becomes a Ba'al Teshuba and begins observing the Misvot, then he may be counted as one of the ten men for a Minyan even though he did not yet learn Hebrew and is thus still unable to recite the prayers.