I'm glad you asked this.
I actually get really angry when I hear people mindlessly badmouthing the Westboro Baptist Church (the "God Hates Fags" people). I even get annoyed when I hear people defend WBC on free-speech grounds, because they always say the same thing: "Sure, I wish we could throw these people in jail, but gosh darn it, we gotta defend free speech!". As if the only reason we let people with odious opinions speak their minds is because of some slippery slope belief that otherwise we ourselves will be censored.
The truth is I'm grateful for WBC's honesty. I don't just reluctantly support WBC's right to free speech, while really wishing they would shut up. I wish they didn't hold the opinions they do, because I think they're wrong, but given that they do hold those opinions, I'm glad they're forthright about it. I don't think anyone would be better off if they kept their hateful views to themselves. Their children are arguably better off being exposed to mainstream culture, knowing how controversial their family's views are, rather than being quietly indoctrinated.
And yeah, I think it really sucks when homophobes pretend to be okay with gay people so they can push their agenda. Whenever you misrepresent your goals, in order to further your goals, you are tricking people. It's tempting, when you have some great revelation you think everyone should share, to think in terms of winning people over. But what if you're wrong? What if your manipulative talking points backfire? Imagine how much worse it would be, if Westboro Baptist Church had come up with some ingeniously subtle way to spread their message, that actually gained converts? Honesty is always the best policy, because it's self-correcting. I'm glad when people I disagree with loudly, publicly speak their opinions, for the same reason I'm glad when people I agree with loudly, publicly speak their opinions: it allows others to judge those opinions on their merits.
I already have a judgment of the opinion of homophobes.
I agree with you on this for slightly different grounds; hatred grows in the dark, and shrivels up in the light. Which is to say, I don't see any particular value in agreement or disagreement or judgment, per se, but I do see value in the ability to offer counterargument. I might not be able to change the opinion of this particular person, but those who might grow to share their opinions otherwise.
(That being said, I wish -my side- would shut up most of the time, particularly on political matters. I have to put up with the reputation their nonsense engenders for us. Sturgeon's Law means free speech is best left for your opponents to hang themselves with.)
Suppose, for a moment, you're a strong proponent of Glim, a fantastic new philosophy of ethics that will maximize truth, happiness, and all things good, just as soon as 51% of the population accepts it as the true way; once it has achieved majority status, careful models in game theory show that Glim proponents will be significantly more prosperous and happy than non-proponents (although everybody will benefit on average, according to its models), and it will take over.
Glim has stalled, however; it's stuck at 49% belief, and a new countermovement, antiGlim, has arisen, claiming that Glim is a corrupt moral system with fatal flaws which will destroy the country if it has its way. Belief is starting to creep down, and those who accepted the ideas as plausible but weren't ready to commit are starting to turn away from the movement.
In response, a senior researcher of Glim ethics has written a scathing condemnation of antiGlim as unpatriotic, evil, and determined to keep the populace in a state of perpetual misery to support its own hegemony. He vehemently denies that there are any flaws in the moral system, and refuses to entertain antiGlim in a public debate.
In response to this, belief creeps slightly up, but acceptance goes into a freefall.
You immediately ascertain that the negativity was worse for the movement than the criticisms; you write a response, and are accused of attacking the tone and ignoring the substance of the arguments. Glim and antiGlim leadership proceed into protracted and nasty arguments, until both are highly marginalized, and ignored by the general public. Belief in Glim continues, but when the leaders of antiGlim and Glim finally arrive on a bitterly agreed upon conclusion - the arguments having centered on an actual error in the original formulations of Glim philosophy, they're unable to either get their remaining supports to cooperate, or to get any of the public to listen. Truth, happiness, and all things good never arise, and things get slightly worse, as a result of the error.
Tone arguments are not necessarily logical errors; they may be invoked by those who agree with the substance of an argument who nevertheless may feel that the argument, as posed, is counterproductive to its intended purpose.
I have stopped recommending Dawkin's work to people who are on the fence about religion. The God Delusion utterly destroyed his effectiveness at convincing people against religion. (In a world in which they couldn't do an internet search on his name, it might not matter; we don't live in that world, and I assume other people are as likely to investigate somebody as I am.) It doesn't even matter whether his facts are right or not, the way he presents them will put most people on the intellectual defensive.
If your purpose is to convince people, it's not enough to have good arguments, or good facts; these things can only work if people are receptive to those arguments and those facts. Your first move is your most important - you must try to make that person receptive. And if somebody levels a tone argument at you, your first consideration should not be "Oh! That's DH2, it's a fallacy, I can disregard what this person has to say!" It should be - why are they leveling a tone argument at you to begin with? Are they disagreeing with you on the basis of your tone, or disagreeing with the tone itself?
Or, in short, the categorical assessment of "Responding to Tone" as either a logical fallacy or a poor argument is incorrect, as it starts from an unfounded assumption that the purpose of a tone response is, in fact, to refute the argument. In the few cases I have seen responses to tone which were utilized against an argument, they were in fact ad-hominems, of the formulation "This person clearly hates [x], and thus can't be expected to have an unbiased perspective." Note that this is a particularly persuasive ad-hominem, particularly for somebody who is looking to rationalize their beliefs against an argument - and that this inoculation against argument is precisely the reason you should, in fact, moderate your tone.