The basic idea has to do with defeaters. A defeater, to put it quite roughly, is something that shows that an argument fails, either by showing that the conclusion is itself false, or that the premises do not support the conclusion, or both. The first kind is called a rebutting defeater, the second an undercutting defeater. There are kinds of defeaters that can be both, and they occasionally are studied, too. In any case, any reference to defeaters is somewhat complicated by the fact that the label 'undercutting defeater' is used ambiguously. To take just an obvious and handy example: Robert Koons's SEP article on defeasible reasoning characterizes undercutting defeaters as giving one "a reason for doubting that the usual relationship between the premises and the conclusion hold in the given case". Michael Sudduth's IEP article on defeaters in epistemology characterizes an undercutting defeater as "a reason (in the broad sense) for no longer believing p, not for believing the negation of p". These are not equivalent. Koons-undercutting is local: an undercutting defeater of this sort is always a defeater for a given premise set in an argument, showing that it does not support the conclusion. Sudduth-undercutting is global: an undercutting defeater of this sort has to undercut all reasons for holding the conclusion, or it's not a reason for doubting the conclusion. A Koons-undercutter is not necessarily a reason to doubt the conclusion -- there may be other arguments, unaffected, that still support the conclusion -- and a single undercutter can only be both a Koons-undercutter or a Sudduth-undercutter if either (1) there is only one available argument supporting the conclusion; or (2) if more than one argument supports the conclusion, it happens to undercut them all simultaneously. The ambiguity seems quite common, and I think traces back to the fact that philosophers from Pollock on have appealed to defeaters both to discuss particular reasons and to discuss justification of belief. One finds similar ambiguities elsewhere; when people talk about suspending judgment, for instance, they usually mean global suspense (I suspend judgment about P) but sometimes slip into talking as if they meant local suspense (I suspend judgment about P to the extent that judgment that P would be based on this argument). I think in many ways it makes more sense (in terms of simplicity, flexibility, utility, etc.) to talk in terms of Koons-undercutting, but the post in question takes a Sudduth-undercutting approach (an undercutting defeater "neutralizes one's evidence for the truth of P" and "gives one a reason to suspend judgment with respect to P"), so we'll go with that. (Although I should say that the appeal to the Morrison paper as providing a paradigmatic example makes this seem a bit murky: as far as I can see, we have Koons-undercutting there, not Sudduth-undercutting. But I may be missing something.)
An important implication of this is that a defeater D may fail to show that P is false, and yet succeed in indicating a live epistemic possibility that's incompatible with the truth of P. In such a case, D succeeds in showing that B ought to suspend judgment about the truth of P, even though D fails to show that B ought to believe that P is false.
"Live epistemic possibility" in this context means that there is a possibility not ruled out by the evidence (consistent with all the premises of the available arguments that support the conclusion) that entails that P (the conclusion) is false. Earlier in the post exapologist had said, "A common form of undercutting defeater for P is a live epistemic possibility (i.e., a scenario that one's evidence can't rule out as false or unjustified) that, if true, entails that P is false." This is certainly not true as it stands, since all probable arguments are by their nature consistent with live epistemic possibilities in this sense -- if a probable argument could rule out all possibilities entailing the falsehood of its conclusion, it would be a demonstrative argument, not a probable one. 'Liveliness' has to be more robust to account for the undercutting defeaters in question; I'm not sure what the best way to revise the claim would be. Nonetheless, the key point here is that if you have shown that a supposed refutation or objection is not a rebutting defeater, it still might be an undercutting defeater. And this is indeed worth noting.
However, how do we get any fallacy out of this? The idea is that "if a person A asserts that P is true, and another person B offers D as a defeater for P, it's not enough for A to show that D fails to show that P is false; A must also show that D fails to neutralize the evidence for P." I dislike talk about what arguers 'must do', since this rarely does justice to real argumentative situations, and because I firmly believe that obligations of argument and dialectical norms must be negotiated and not merely dictated by fiat, but we can rephrase this pretty easily: if a person A asserts that P is true, and another person B offers D as a defeater for P, if A shows only that D is not a rebutting defeater, it may nonetheless still be a defeater, namely, an undercutting defeater. On the basis of this, we get to the fallacy in the following way
The preceding discussion reveals a dialectical norm: in dialectical contexts of the sort sketched above, a person in A's position must not only show that (i) D fails as a rebutting defeater for P, but also that (ii) D fails as an undercutting defeater for P. And to assume that A discharges their dialectical obligations in offering justification for P to B in such contexts by accomplishing (i) alone is to commit a certain sort of dialectical fallacy.
In fact, this is not true, and part of the reason this is not true is that we are dealing with Sudduth-undercutting, which is global and requires cutting off a conclusion from the entire field of evidence apparently supporting it. If I make a claim, like "Milk is full of nutrients," and you give some supposed defeater or other, and I show that your supposed defeater does not show that my claim is false, there is absolutely nothing unreasonable about leaving it at that. It is true that it could still possibly be an undercutting defeater, but merely failing to be completely thorough is not a 'fallacy'. No one thinks that everyone has a rational responsibility to show that an objection not only doesn't prove their conclusion wrong but also doesn't eliminate their entire field of evidence. For one thing, actual objections that successfully eliminate every available line of evidence for a conclusion supported by several different lines of evidence are going to be rare. In general people will assume -- and in general it is a reasonable thing to assume -- that if you have lots of lines of evidence for X, no single undercutting defeater will defeat them all, even if it manages to be successful for defeating some. To take an extreme example: if someone proposed a possible defeater to Darwin on natural selection, and he showed that it did not prove his conclusions wrong but left it at that, would anyone seriously blame him for not also showing that it wasn't a Sudduth-undercutter given that Darwin's argument for natural selection is a book-length argument with dozens of distinct lines of argument? Of course not. For another, Sudduth-undercutting is difficult to give success conditions for, much harder than Koons-undercutting, because people can have honest disagreements about what is to be counted in the field of evidence. What do we have to do to give reasonable certainty that an apparent defeater really does cut off the conclusion from every available line of evidence, so that rather than believe the conclusion we should simply suspend judgment? In Koons-undercutting we have specific arguments in view, and we only have to show that the defeater cuts off the conclusion from these specific arguments (but we still leave open the possibility that we should not suspend judgment about the conclusion for some other reason that hasn't been considered); in Sudduth-undercutting, however, we have to make sure that the defeater cuts off the conclusion from every argument that might currently be put on the table for it. The latter is such a high standard we can only occasionally meet it uncontroversially, and it is difficult to see why anyone should regard it as having definitely been met until someone actually shows that it has been.
It seems to me, then, that (1) it would be more sensible to put this in terms of Koons-undercutting rather than Sudduth-undercutting, and thus do away with the whole useless talk about suspension of judgment, which is rarely even a serious option on the table except with very weakly supported positions or very powerful skeptical objections; and (2) regardless, it is obviously not a 'fallacy' of any sort, but just a way that an argument can fail to be a complete answer, which is an entirely different sort of argumentative weakness.