I'm going to tell my friend to give away his dog
Why would you want your friend to get rid of his dog?
Because he asked me to take care of his dog when we went out.
Your friend has a loving and committed relationship to his dog, and its hurtful for you to even suggest that he give that up.  You should be willing to help your friend take care of the dog temporarily, because it is only a small inconvenience, and would mean a lot to your friend.
I'm not a dog owner and I don't have time to take care of a dog.
It would be kind of you to help your friend out with his dog. But you're not required to do that either. Just tell him that you can't, and then he'll find someone else to do it.