I'm not going to give my friend money for treatment of his dying cat
Is there a reason why you wouldn't want to help them out?
It's not my cat therefore I can't help him look after it.
It's fine to not want to give your money to someone else. It's your money and you have the right to not loan it out to people you don't want to.
I'm afraid my friend will use my money to do something else instead of using it for his cat.
If you're suspicious of your friend's intention then definitely you're doing the right thing by not giving him any money.