Is it "unloving" to warn people about eternal damnation for their unrepented sins?

No, it is the most loving thing we can do. We are risking getting attacked by this but this is a command from God to warn the wicked. When we are saved, we want to help others to get saved too because otherwise they will burn in hell. I often recommend that people go to a revival because that way they can get inspired and hear the word powerfully, and get a real chance to trust in the Lord and become a child of God. We are under condemnation until we repent and trust the Lord with our lives to be our Lord and Savior.