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.