Killing old and sick animals

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.”