I'm toying with an indicator theory of why this is wrong. That is, perhaps "love the sinner, hate the sin" isn't wrong because of its literal meaning, but because the people who tend to espouse such attitudes tend to be the problem people.
And yes, "love the sinner, hate the sin" definitely correlates with negative attitudes towards homosexuality, that's not just an impression. According to a paper in the International Journal for the Psychology of Religion:
I wondered if the “hate sin, love the sinner” responses of persons scoring in the top quartile of the Religious Fundamentalism scale would be reflected in their attitudes toward homosexuals. They were not. Most “High Fundamentalists” agreed—strongly in fact—that one should hate sin but love the sinner. But they nearly proved significantly more rejecting of homosexuals (M of 51.8) than did the few High Fundamentalists who disagreed with hating sin but loving the sinner (M of 37.7; t = 1.70, p < .10). And, of course, they rejected homosexuals much more than the other three quarters of the sample. They may believe in loving the sinner, but they also believe much more that homosexuals should be discriminated against and even thrown into jail. [emphasis mine]Here's another example where I'm quite sure that the indicator theory is the correct one: "Some of my best friends are gay." There's obviously nothing wrong with having gay best friends. Knowing gay people personally has a greater positive effect on one's attitude towards gay people than nearly anything else. The problem with saying, "some of my best friends are gay," is that it's the flimsy defense people give when they're being defensive, and the kind of people who need to get defensive tend to be the problem people.
Another example: using "gay" as a noun. In many other identity categories (eg atheist, asexual, lesbian, vegetarian, Asian, American) people hardly think about whether they're using them as nouns or adjectives. I think the reason we consider "The gays" to be wrong is because for whatever reason that usage mostly comes from the problem people. The more informed people know that, and therefore avoid using "gay" as a noun.
Of course, just because "love the sinner, hate the sin" is an indicator of problems does not mean that its literal meaning does not also have problems. The problem with its literal meaning is that it's espousing a friendly attitude while simultaneously admitting terrible underlying beliefs. Respecting others in the face of disagreement is a virtue, but it is outweighed by the fact that the person believes in morality derived from a contemporary interpretation of a text that comes from nearly two millenia ago. When people have such stupid reasons to hate something (be it people or behavior), one is tempted to more than just disagree with them, but to hold them morally accountable for such wretched views.
To further explore what is or isn't wrong with the statement, I always find it useful to try to think of similar cases where my sympathies lie in the other direction. For example, if someone tells me that they're going to see their acupuncturist, I might tell them that I think this is wrong because acupuncture is bunk and a waste of money. But I wouldn't dislike the person for this. So in some sense, I'm
Another example: I am an omnivore, but several of my friends are vegetarian or pesco-vegetarian. If they tell me that they think eating meat is morally wrong, but don't let this interfere with their friendship, that's okay with me. Vegetarians have their reasons, and these are cool to talk about on occasion. In some sense, vegetarians love the sinners (their omnivore friends), and hate the sin (eating meat). What's different?