Damis lays it all on the line in Act 1, Scene 1, when he tells Madame Pernelle that Tartuffe's "every action makes me seeth and tremble/ with helpless, anger, and I have no doubt that he and I will shortly have it out" (1.1.21). He spends the rest of the play looking for the opportunity to pick a fight with Tartuffe, but he never gets to throw any punches. His anger does get everyone in a pickle, though, when he insists on jumping out of the closet, confronting Tartuffe face to face, and telling Orgon the truth of the matter. In this case, Damis's insistence on truthfulness only ends up getting him disinherited.
His eagerness to settle things with Tartuffe does have a clear origin, though: he wants to marry Valère's sister. As he tells Cléante, his "hopes…will miscarry" (1.3.3) if Tartuffe weds Mariane instead. So, basically, he's looking out for his dad, his sister and, well, himself. There's nothing wrong with that. His strategy just seems to be, uh, less than effective.