Is taking money from a discarded shirt in an institution and keeping it for over four years out of dire need considered a sin, and what is the religious obligation concerning this money given the owner's wealth?

Taking money without the owner's permission and using it is forbidden. Repentance is obligatory, and what was taken must be returned to its owner unless he absolves the taker. The fact that the taker is poor and the owner is rich does not permit taking it, due to the Prophet's (peace and blessings be upon him) sayings: "The wealth of a Muslim person is not lawful except with his willing consent," and "On the hand is what it took until it returns it."