Psyche has a track record of this trope: twice after she loses Cupid, and once after each of the last three impossible tasks Venus orders. Considering the raw deals she has got, it is hard to blame her.
In most versions of the story of Oedipus, Jocasta hangs herself once she realizes the man she has married is her son and her husband's killer. Oedipus, however, punishes himself in a far-more long-lasting way.
Heracles' third wife Deianira hangs herself in grief when she realizes that she's been tricked into poisoning his robe, which eventually leads to the hero's death.
Some versions of Ariadne's myth claim that she hanged herself, when she realized Theseus left her on Naxos to die.
In one version of the story of Arachne, Athena's intended punishment is a curse intended to teach humility and respect; it works too well, filling her with guilt and causing her to hang herself. Athena grants a small consolation by turning her into a spider.
One version of the story of Jason and the Argonauts says that Jason ultimately, after breaking his vow to Medea and thus losing Hera's favor and becoming broke and destitute, tried to hang himself from the stern of the Argo, only for it to break off and crush him. (And that's the compassionate ending. The crueler one says he went to it to commiserate, saying it was his only friend; thenit fell on him.
Judas hangs himself from a tree after betraying Jesus, according to Matthew's Gospel, although this is contradicted by Acts of the Apostles.
King Saul at the end of 1st Samuel, when he was badly wounded in the battle with the Philistines, tells his armor-bearer to slay him so that his enemies wouldn't make sport of him, but since his armor-bearer refused, Saul falls upon his own sword and dies, and so also does his armor-bearer. An Amalekite who goes around Robbing the Dead comes to King David with the personal effects of King Saul and tells the story of him having to slay Saul when he was badly wounded in battle, hoping to get a reward for the cool loot he had taken. However, the only reward the Amalekite gets is to be killed by King David himself, since according to David the Amalekite testified by his own mouth that he had killed "the Lord's anointed".
Ahithophel, an adviser to King David's son Absalom during the coup he staged against his own father in 2nd Samuel, kills himself after Absalom chooses to accept Hushai's advice as to what to do concerning King David over his own advice.