There are basically three tests that an objection against divine command theories needs to pass in order to count.
(1) Adequacy: Obviously any objection against an argument that is worth taking seriously has to get the argument right, or at least reasonably close to right, and avoid arguing instead against a simplistic caricature.
(2) Restriction: Divine command theories are a form of moral positivism and, structurally speaking, one of their natural advantages is that by being so they make possible a highly unified theory of obligation. We know that some standards obligate because they are promulgated by sanctioning authorities; traffic laws, for instance. It is possible to hold some form of the view that all laws are like this (or else become laws derivatively on the basis of laws that are like this); this is positivism about laws. If you hold that all obligations are the same, this is positivism about obligation in general, or what we can call moral positivism. Divine command theory has the advantage over other moral positivisms that it posits a truly universal and ultimate sanctioning authority, and therefore is capable of giving a truly general theory of obligation that can handle very robust kinds of moral obligation. So this is a strong point for divine command theories in general; if you have an objection against divine command theories, it had better be able to handle divine command at its strong point. And the test is seeing what happens when you restrict the objection to ordinary human laws. If, so restricted, someone could deny that there are any laws on the basis of your objection, it's a bad objection to divine command theory in particular; you have a problem with obligation, not divine command. If, so restricted, there's an obvious answer to your objection for human-made obligations, then it fails as an objection against divine command theory, because either you can scale that answer up due to divine command theory's general theory of obligation, or, if your objection only survives by positing some principle of difference between the two cases, your principle of difference is both doing all the work and begging the question against divine command theory.
(3) Warburton: It need not strictly be Warburton in particular, but Warburton is the handiest, because William Warburton (1698-1779) developed a well-rounded and intensely defended version of divine command theory. I've found by long experience that people proposing objections that are supposed to apply to divine command theory in general are often outmaneuvered in advance by Warburton; he's anticipated their objection, or something like it, and his version of divine command theory is immune to it. Of course a lot of times objectors are not specifically thinking of Warburton at all; Wielenberg, for instance, is looking only at recent work on divine command theory. But passing the Warburton Test does not require that you directly engage with Warburton; it requires that your argument not be something that Warburton had obviously outthought centuries before you came up with it. If you fail the Warburton Test, it's still conceivable that your argument might work against this or that particular version, but if you claim that you have an objection against divine command theory in general and you can't handle Warburton, your claim is false. And it at least raises the question whether you really understand divine command theories well enough to be proposing general objections to them. So a minimally reasonable objection should at least be something that the basics of Warburton's account don't already rule out as at least a worry.
How does Wielenberg's objection do in passing the tests? Not bad.
Wielenberg's argument passes the Adequacy Test adequately. DCT, he says, holds that
an act is morally obligatory just in case God commands it, morally wrong just in case God forbids it, and (merely) morally permissible just in case God neither commands nor forbids it – and it is God’s commanding, forbidding, or doing neither that in some sense grounds the moral statuses of actions.
This is good -- it avoids many of the obvious failures you find, like not recognizing that DCT is a theory of obligation, not a general theory of goodness. I could quibble with the last clause; it's not uncommon for God's sanctioning, rather than commanding as such, to be what makes things obligatory, but given that sanctioning is in these cases is tied to commanding, I think the "in some sense" is a qualification that works just fine. Wielenberg goes on to add that obligations have to be communicated; he calls this a "distinctive feature" of DCT, which I don't quite understand, since it's not something that distinguishes DCT from other theories of obligation, but a principle of obligation that is extremely widely held, nor does it seem to be held by divine command theorists in any especially distinctive way. But it is widely held, it's very difficult to build a coherent theory of obligation that holds we can be obligated by standards that are in no way communicated to us at all, and even if it is possible to build a divine command theory that denied it -- I think it probably is possible -- I don't know any actual divine command theorist who definitely denies it. So far, so good. On both of the key points on which Wielenberg focuses, he avoids constructing a straw man; it's still perhaps possible to have theory that is obviously DCT that doesn't quite fit these assumptions as given, but Wielenberg seems to capture most of the field so far. But simply this is not enough; given the defining feature noted above, he needs to have a reasonably acceptable understanding of command and the conditions of its communication. He chooses the following:
(R) God commands person S to do act A only if S is capable of recognizing the requirement to do A as being extremely authoritative and as having imperative force.
It's a bit tougher to assess how general this would be for an assumption in DCT. Obviously the "extremely" is a potential issue. Wielenberg is getting this from Adams, but I'm not sure we should be taking Adams on this point as expressing something DCT in general has to hold. This ties up with worries raised by the Restriction Test: it's implausible that commands in general have to be recognizable by the commanded as "extremely" authoritative, and so, while God's commands can be assumed to be extremely authoritative, it's not clear that we can't be commanded by God even if under the circumstances we can only recognize that the requirement is authoritative in some way. At my college, which has multiple campuses, Campus Managers have the authority to impose rules on the campus; they do this generally by posting signs. I don't even know what it would mean to recognize campus rules as "extremely" authoritative. But when the sign is posted, I am capable of recognizing that it has some kind of authority, and therefore it could be said to be commanded. I have been told, in an imperative form, by some kind of relevant authority, and I am in principle capable of recognizing it. And if the campus manager doesn't need more than that to obligate me, it's not clear why God would need more.
However, as far as I can see, Wielenberg makes no real use of the 'extremely'. It's completely otiose; when he describes (R) more informally, he never uses the 'extremely', just the idea that you can recognize it as authoritative. So despite the potential worry if someone were to use Wielenberg's description as a general characterization, I think Wielenberg passes the Adequacy Test in practice.
But the Restriction Test is a tougher test. From the abstract, I was worried that Wielenberg would not even address the issues required for it, but I was happily relieved; he uses examples based on signs posted by relevant authorities, so he's at least sometimes bringing the discussion back to the kinds of things you need to consider in order to pass the Restriction Test. But really to see how well it fares, we need to look at the actual steps of the argument, which are (in my paraphrase):
(1) There are psychopaths who cannot grasp the authority and force of moral commands.
(2) Therefore there are psychopaths whom God has not commanded. (from 1 and R)
(3) Therefore, if DCT is true, they do not have moral obligations (from 2 and the defining idea of DCT)
(4) But psychopaths do have moral obligations.
(5) Therefore DCT is false.
I do not think that (1) is as straightforwardly true as Wielenberg wants it to be; to get (2), it needs to use terms in the same sense in which they are used in (R), but (R) just requires that people be able to recognize authority and imperative force, and there are no restrictions on what counts as long as it's a recognition. But Wielenberg's authorities explicitly say that psychopaths can know what the rules are, so they can recognize that something has imperative force, and their capacity to manipulate suggests that they can identify what is supposed to be authoritative. What they don't do is feel it or apply it to themselves, but neither of these is explicitly required by (R), and there's a serious danger of equivocating between 'recognizes authority and imperative force' in a sense actually required by DCT and 'recognizes authority and imperative force' in any number of stricter senses. However, right now we're look at how the argument succeeds in terms of the tests, and not at the truth of the premises. The real problem at hand is this: under the Restriction Test, bringing the argument down to the domain of human-commanded obligations, many of which we know to get their force from command (and we certainly cannot merely assume otherwise when arguing against DCT in general), (3) implies that psychopaths cannot be obligated by human-made laws and rules. To pass the Restriction Test we don't have to prove that this is true. But it has to be reasonably defensible.
We clearly can make rules that we expect psychopaths to follow. We can make it known to psychopaths that they must follow them. We can punish them when they don't. That suffices to obligate a dog. Why does that not suffice to obligate a psychopath?
Wielenberg does not in any way directly address this. But because he brings it back to the posted sign examples, he does address something in the vicinity. And I think it commits him to saying that the bare fact that they are psychopaths means that they are not obligated to follow laws or rules that are based on commands, including our own. Suppose, he says, a city government makes a bunch of laws requiring stops at intersections, but they fail to do what they should do to promulgate this properly; instead of posting the correct STOP signs, they just put up construction paper signs with STOP written on them in crayon. This is obviously a problem if they didn't tell anyone about this. But suppose they did tell everyone about this except that somehow they missed you. Then, Wielenberg says, he doesn't think that this is enough to generate an obligation for you in particular; people who were told are obligated. You are not. If this is right, Wielenberg is perhaps committed to biting the bullet: people need to be able to feel the authority and imperative force of our requirements for there to be any obligation, so it's impossible for us to command a psychopath.
This, I think, is enough for passing the Restriction Test, although I am baffled by this answer. The City of Austin almost never directly informs me of anything; and as the City Council in Austin is famous for its incompetence, I really wouldn't put it past them to do something like put up confusing or misleading signage, and, while greatly to be pitied, I'm not sure why I wouldn't still count as being obligated. Laws require promulagation; they don't generally require promulgation directly to me for me to bound by them. If it were literally impossible for me to know it, that would be one thing, but if other people can tell me that it's a law, the fact that I couldn't determine it directly myself would not be counted as an excuse. But Wielenberg's comment is enough to think that we could restrict to a lesser domain and still make sense of things.
And that leaves the Warburton Test. On Warburton's view, obligation depends on command; he thinks people who recognize obligations but not obligator are incoherent. But I think he does hold that they can be right about their obligations; this is possible because they live in a society with people who do think rightly about obligations, at least in a general way, and God has given us two things (moral feeling and the ability to recognize relations of perfection) that, while not enough to give us obligations, under normal circumstances act as guardrails to keep us from wandering too far from what God has actually obligated. So can we still formulate something like Wielenberg's argument in something like a Warburtonian context. (We don't need to prove that Wielenberg's psychopath argument works against Warburton for it to pass the test; we just need to have reason to think that (1) it or something reasonably similar can still make sense as a worry in that context and (2) Warburton has no immediate and obvious answer to it, in the sense that the structure of the theory itself automatically provides an answer.) Psychopaths are interesting in this context because they seem by definition to be defective in moral sentiment, so they are lacking one moral safety mechanism that prevents most people most of the time from doing horrible things. But Warburton doesn't think that our obligations depend on moral sentiment; it just helps keep us from deviating too far. And there's no obvious reason why psychopaths couldn't recognize relations of perfection, which is a purely rational matter. So is being in a society that has received communication from the obligator sufficient?
And Wielenberg, of course, has a response to this -- it is what he was directly addressing in the STOP sign example, in a slightly different form, but the difference doesn't really matter. And while Warburton would obviously not agree with Wielenberg's assessment, assuming Warburton's theory doesn't make it a nonsensical worry.
So Wielenberg's argument seems to pass all three tests. There are quibbles that can be made, and passing the tests just means that it's at least a minimally acceptable objection to divine command theory. But it has been a long time since I've come across an objection that was set up in such a way that, despite not being deliberately made to pass all three tests, actually had more or less everything it needed to do so. That's nicely done. I think the premises need a lot more support than Wielenberg had the chance to provide in a single paper, and I am skeptical of most of them. But it's an argument worth considering.