Killing old and sick animals
Question

                is it permissible to put an animal to sleep (euthanasia) if the animal is old and in pain?
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.”