Gay rights advocates often claim that gay rights are simply one more type of civil right; or conversely, homophobia is the same kind of thing as racism. To think homosexuality is immoral is as illogical as saying that being a certain race or ethnicity is immoral.
But what exactly do some say is immoral? Homosexuality? Homosexuals? Or homosexual acts? Traditionally, the Judeo-Christian claim is that only the last of these is immoral. Homosexuality is the ongoing temptation to engage in homosexual acts, but temptation is not immoral even if what you're being tempted to do may be. Moreover, the Christian is commanded not to condemn the person (the homosexual) but only the act. This is the meaning of the proverb "Love the sinner, hate the sin." (Admittedly this proverb is not in the Bible, but it summarizes the biblical position well.) Both of these are in some sense dependent on the third category, homosexual acts or behavior. To say someone is homosexual but is not tempted to perform homosexual acts is a contradiction in terms. Similarly to refer to homosexuality without any concept of same-sex attraction simply doesn't make any sense. The temptation to perform homosexual acts is what "homosexuality" means.
Now the problem with claiming homophobia is the same sort of thing as racism is that these categories do not transfer to race or ethnicity. This is most obvious with the third category. We all know what homosexual acts are: they are sexual acts between members of the same gender. They are not merely acts performed by homosexuals; when a homosexual washes his car, he is not engaged in a homosexual act. A homosexual act is one which defines the act as homosexual inherently, that is, by its very nature.
If homophobia were the same sort of thing as racism, there should be corresponding acts or behavior that are particular to different races. Just as there are homosexual acts, so there should be Chinese acts, or Hispanic acts, or white or black acts. This is obviously absurd. Therefore, homosexuality is not the same sort of thing as race or ethnicity. As I argued in this post, homosexual behavior involves an element that is simply not present in race: namely, behavior.
The first and second categories are problematic as well, since they are dependent on the third. A homosexual is someone who is tempted to perform homosexual acts. But a white person is not someone who is tempted to perform "white" acts, since there are no such acts. To say someone is homosexual but is not tempted to engage in homosexual acts is to redefine the word "homosexual." A homosexual is a person who is tempted to perform homosexual acts by definition. Similarly for the first category: the equivalent of homosexuality would be "whiteness" or "blackness" or whatever. But "whiteness" is not the temptation to perform "white" acts because (again) there are no such acts. Thus to treat homosexuality as the same sort of thing as race or ethnicity -- and homophobia as the same sort of thing as racism -- is simply invalid.