Praise be to Allah.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen was asked this question and he said:
If an animal gets sick, if it is an animal whose meat it is not permissible to eat, and there is no hope that it will recover, then there is no sin on you if you kill it, because if it is kept alive it will become a burden on you which may be a waste of your money, as you will have to spend on it and this spending will be a waste of your money. Keeping it alive until it dies without giving it food and drink is haraam, because the Prophet (peace and blessings of Allaah be upon him) said: “A women went to Hell because of a cat which she detained. She did not feed it when she detained it, and she did not let it free to eat of the vermin of the earth.” But if it is an animal whose meat may be eaten, and it reaches a state where you cannot benefit from it or give it to someone else who can benefit from it, then the ruling is the same as that concerning an animal whose meat is haraam, i.e., it is permissible to kill it, whether you slaughter it or kill it with bullets. Do whatever is easiest on the animal, because the Prophet (peace and blessings of Allaah be upon him) said: “If you kill, kill well, and if you slaughter, slaughter well. Let each one of you sharpen his blade and let him spare suffering to the animal he slaughters.”