Overall, male against female discrimination contributes to Okonkwo's tragedy, but I don't think it contributes to the collapse of the tribe. White against black racism and Christian monotheism against tribal polytheism contribute more toward the collective unraveling of culture.
In Part I of Things Fall Apart, the Igbo tribe is segregated in terms of gender roles. Men farm for yams; women cultivate cassavas and beans. Men take part in the wrestling and masked dancing; the women prepare for these feasts.
Whether this is gender discrimination or sexism depends on which culture is framing the question. Our independent modern culture, which champions integrated gender roles and feminism, would certainly call this male prejudice against women. Specifically, Okonkwo has rigid roles that he feels he should play, as well as his wives and male and female children. For example, Nwoye acts a bit feminine and Enzimna acts a bit masculine, but Okonkwo makes sure that neither crosses over into the gender duties of the other. These roles contribute to Okonkwo's fear of being weak, which--in turn--leads to his exile and eventual suicide. These differences alone, though, do not cause the group to fall apart. In short, they tolerate what we call sexism.
Religion is a major unraveling agent in Parts II and III. The white Christian colonists feel that the Igbo are polytheistic and, therefore, pagan. Little do the British know that the tribe has one major god and many other gods doing his service. Conversely, the tribe has little religious prejudice against the Christians. At first, the Igbo largely ignore the Christians because they have set up camp in the evil forest and, as a result, the Igbo feel the missionaries are going to be obliterated. The survival of the church and the converts obtained are a major contributing factor in the tribal split. The Christians expose the tribe's practice of killing twins and cruelty to outcasts.
Racially, there is some white against black racism, but it is not as developed as the religious differences.