Why does Hale quit the court?
In addition to the reasons outlined in previous answers, Hale is no longer comfortable signing death warrants for individuals convicted by this court. He says, in Act Three, that he has signed some seventy-two warrants, and he cannot sign more if he has any shred of doubt about the validity of these convictions. When Danforth is unwilling to allow John Proctor and Giles Corey to leave and return with a lawyer to help them present their evidence (and Mary Warren's deposition) to the court, Hale becomes very concerned that this court and its judges are actually corrupt. He believes, by the end of the act, that Mary Warren has gone wild and that the court should not be willing to accept her testimony against Proctor. However, Danforth's unequivocal trust in her words, as long as they support the other girls (who he believes are serving as God's voice in Salem), is too much for Hale, and he quits the court.
Hale quits the court in Act 3 because he no longer believes that the girls are telling the truth and feels that the judges are blind to the lies that they are telling. John Proctor is not successful in convincing the court of his affair with Abigail; since Hale knows what kind of a person Proctor is, he now truly believes that he is being truthful and that the girls are fabricating the entire story. Therefore, he quits the court. In Act 4 Hale continues to play a part in the play when he tries to convince certain characters to confess to witchcraft ONLY because he does not want to see them get hanged.