Are commas required ? I haven't used the commas.
I consider the first comma optional in the revision I suggested even though "even though"-clauses usually are set off by commas. If "in spite of" or "despite" is used instead of "even though," the comma is even more optional. In the earlier responses in this thread, the ending part of the original sentence was ignored. Without that ending part, the absence of a comma makes sense: "I have to go to school in spite of (my) being tired." That expresses a single thought. With the ending part included, however, the logic of the sentence changes, and that affects comma choices. We want the "because"-clause to give the reason for the speaker's having to go to school, not for his being tired:
I have to go to school because I have exams.
? I am tired because I have exams.
To ensure that the "because"-clause will be read as relating to the speaker's having to go to school rather than to his being tired, we can make the "in spite of" phrase parenthetical by setting it off with a comma at each end. That the speaker is tired provides an extra, noteworthy detail, but it does not affect the basic thought, which is that the speaker has to go to school because he has exams.
I have to go to school, in spite of (my) being tired, because I have exams.