All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
In principle, it is not permissible to kill a domestic animal without a sound reason, like for the purpose of eating it (if it is eatable) and the like. If the purpose of killing it is for relieving it from pain, then this is being merciful to it. This applies when its disease is very serious and it is very expensive to treat it, in such a case it is permissible to kill it. Rather, some jurists stated that it is desirable to do so, which is the predominant opinion, because it concedes with the general Islamic principles of being merciful to animals.
For more benefit, please refer to Fatwa 83830.