Like I said, you've probably heard something like that before. But something I've heard a lot more recently is the idea that Jesus would never hurt someone's feelings. So if you're willing to call X a sin even though it makes X-lovers feel bad, you're being un-Christlike. This is supposed to trump the argument without needing to address it - since, you know, the argument is fairly obvious and universally acknowledged in any other sphere of life, and to try to counter it would be plainly silly. So instead we get the ultimate trump card - Jesus wouldn't do that, so why do you?
The big problem with this objection is that it's demonstrably untrue. When someone is in danger of hell, Jesus is all too willing to hurt their feelings, to expose the sins they're the most ashamed of, and to crush the idols that keep them from faith and repentance.
Consider the twin evangelistic encounters of John 3 and John 4. In the first, Jesus confronts Nicodemus, a teacher/scholar and one of the most educated and powerful men in Israel, and essentially calls him a worldly-minded ignoramus. In the second, he cuts through the Samaritan woman's political/social/theological banter by going straight to confronting her promiscuity. Nicodemus was no doubt crushed to have his ignorance exposed, and the Samaritan woman was obviously ashamed that he knew the truth about her. In both cases, it was necessary to destroy their idols and walls, because they were keeping them from seeing Jesus and being saved. So he did what was necessary to get their attention, despite how it would make them feel, because the stakes were way too high to do otherwise.
Even more directly, consider the astonishing encounter in Luke 11. First, Jesus unleashes a series of condemnations against the Pharisees. A lawyer heard what Jesus said, and thought it maybe hit a little too close to him and his kind. He pointed this out to Jesus: "Teacher, in saying these things you insult us also.” Now surely, if our objector is correct, Jesus would stop to explain that he didn't want to offend anyone, maybe even apologize for any offense taken. But what did Jesus actually say? “Woe to you lawyers also!" And then he launched into a series of condemnations of them as well. Far from backing off to make sure nobody's feelings were hurt, he went out of his way to make sure they didn't miss the insult!
Were there times that Jesus cared about how people felt? Sure. When someone was hurting, such as Mary when her brother Lazarus died, he wept with her and comforted her. And that's when we should care about feelings as well. People who are hurting are to be comforted and helped. But defiant rebellion against God is not to be pampered - it is to be confronted and repented of, lest it last until it is ultimately crushed. That is what Jesus did, and so must we, if we want to love as Jesus loved.