Islam, the true religion of Allah, cares about the well-being of all creatures. Therefore, animals should be treated kindly and mercifully. If a pet cat becomes sick it should be treated.
Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada states: “If your cat is sick and it can be treated you must do so if it is within your means and ability. Simply allowing a cat to be put down because it is sick is not allowed in Islam; should you do so, you will be responsible before Allah for causing unnecessary suffering to an animal.
If, however, your cat is too sick to recover and there is no hope of it ever being able to lead a normal life, or you do not have the means and resources to obtain adequate treatment for it, then you must hand it over to the humane society. It is up to them to decide what to do with the cat. If they decide to put it down because of not being able to find a home, or because of their decision that such an animal cannot be cared for or set free, it is permitted to do so according to the principles of the Shari`ah. It is generally agreed by scholars that if leaving certain animals free becomes a cause of genuine concern for the safety of society, it is permitted in Shari`ah to put them down as a lesser of the two evils.”