Do you believe that God has moral obligations? Why or why not?
Obviously this question will depend very much on one's theory of obligation (by which I mean not a theory yielding specific obligations but a theory of what obligations are and why we have any at all), and as it happens there are a very great many different theories of obligation. Because of this, everything I can manage to say in a single post will be rough and merely sketched-out. Interestingly enough, however, on most commonly accepted theories of obligation it's difficult to make sense of the claim that God has obligations in a non-metaphorical sense of the term, at least any that we could know about. (Of course, anything can have obligations in some metaphorical sense of the term, and the question in such a case would be what that particular metaphor is trying to convey; and while one could hold that God has obligations but we don't know anything about what they are, that is, for practical purposes, not really much different from saying that he has no obligations of any sort we can recognize.)
Roughly speaking, there are three major families of theories of obligation (not strictly exhaustive, although genuine alternatives seem fairly rare, and not mutually exclusive, although full-fledged hybrids seem fairly rare): those based on the claim that there is some kind of obligating authority (two major kinds of these); and those based on the claim that there is some kind of obligating sentiment. In the first obligation tends to be seen as a sort of legislation; the second obligation tends to be seen as a sort of drive or impulse (at least in the mentally healthy).
The sentiment-group tends not to have strong reasons to regard any obligations universal: if obligation is based on sentiment, then it depends at the very least on the nature of what has the obligation. Human beings can share obligations, because healthy and thriving human beings have the same nature, and so the same basic impulses and drives. If, for instance, as Hume posited, there are particularly moral sentiments, then these sentiments can obligate. As he says in talking about promises (Treatise of Human Nature, Book III, Part V, Section V):
All morality depends upon our sentiments; and when any action, or quality of the mind, pleases us after a certain manner, we say it is virtuous; and when the neglect, or nonperformance of it, displeases us after a like manner, we say that we lie under an obligation to perform it.
As Hume himself recognized, however, an implication of this is that any clear understanding of obligations that we might have is confined to the human species as it exists now. If human nature were to change fundamentally, our obligations would change. And if we do not know the natural sentiments of a being, say, God, we do not know what obligations, if any, that being might have. This is a point Hume explicitly makes in a letter to Francis Hutcheson, whom he criticizes for ignoring it. Obviously not every sentiment-based account of obligation need be Humean; but Hume's problem will arise as a challenge to anyone who accepts such an account and holds both that God has obligations and that we can know anything about them. What one would seemingly have to do to answer the challenge is argue that God has to have sentiments sufficiently analogous to our own to allow us to talk about them. But this is not really consistent with most accounts of how we know anything about God.
Hume and a few others aside, most people ground obligation on authority rather than sentiment. They can be divided into two groups: those who hold that all obligating authority is extrinsic to the obligation itself and those who hold that in at least some cases the obligation itself can carry intrinsic obligating authority. The first group is arguably the largest family of theories of obligations, so let's take it first.
Extrinsic-authority theories of obligation can be of almost as many types as there are candidates for sources of authority, but in a sense they all depend on the imposition of will by means of some kind of sanction. One of the primary advantages of pinning obligation itself on an extrinsic authority is that pretty much everyone agrees that at least some extrinsic authorities can impose at least some obligations -- most people agree that the legislature can impose obligations on people, for instance. So the extrinsic-authority theorist is taking a phenomenon we all know well and generalizing it. Two versions in particular have been extraordinarily popular throughout history: divine command theories of obligation and social demand theories of obligations.
In divine command theories, say that of William Warburton in the Divine Legation of Moses, all obligation presupposes an extrinsic obligating authority. This need not strictly be God, but the obligations imposed by an authority are limited to the extent of power that authority wields: to find any universal obligations, one has to go to an authority with genuinely universal power. And obviously, God's the candidate who springs to mind for most people. On such a view God can only have obligations if He can obligate Himself. But as Warburton (if I recall correctly) argues at some length, the sense in which anyone can obligate themselves is a figurative sense of the word: if an obligation depends on your will, and there is no will higher countervailing you, then you can abolish the obligation as easily as you can impose it -- and an obligation that you can do away with as you please is not an obligation in the strict sense. This raises a serious challenge to anyone who holds a divine command theory of obligation and wants to say that God has obligations.
The other very popular extrinsic-authority theory is that obligation is established by the demands of society. Obviously when talking about divine obligations, the question immediately becomes, what society would have the ability to impose sanctions on God? And it's at the very least not clear, unless we are polytheists, that it is even coherent to talk about a society that can impose sanctions on God. But even if it is, how do we know anything about such a society? This, at least, would be a significant challenge for anyone who holds a social demand theory of obligation and wants to say that God has obligations we can know about.
So it would seem that the only hope for someone who wants to say that God has recognizable obligations is some kind of intrinsic-authority theory, in which at least some obligations obligate simply by being what they are. And indeed all the intellectually respectable accounts of divine obligations I can think of are intrinsic-authority theories; in general, they are the only theories that can easily handle the problem of how we can know anything about divine obligations (assuming that they present an account on which God could have obligations); although this is not strictly an advantage all of them share. The trick, then, would be to give an account of obligations in which it would be clear both (1) that this is a plausible account of the authority carried by obligations; and (2) that this account implies that even God would have obligations.
It is important to keep in mind, however, that not all intrinsic-authority theories of obligation do meet this second condition. An initially promising account of obligation for the believer in divine obligations, for instance, would be that of Kant, who holds that human obligations arise from features of human reason that necessarily and universally apply to anything that can genuinely be called rational. But Kant actually denies that God has any obligations. To have an obligation, there must be some constraint on will: that is, obligation or duty does not arise merely from the necessity of reason but from the fact that the necessity of reason would, if applied, restrict desires we have that would lead us away from doing what reason requires. But, says Kant, if there were a holy will, this latter condition would never be met. If God did have duties, we know what they would be, because God has divine reason, and duties in the strictest and most basic sense are necessary constraints of reason; likewise, on Kant's view we know something about God's moral life because God always acts according to divine reason. But nothing constrains God to do so: God, so to speak, acts as if he did have obligations, but God in fact does spontaneously and without obligation what we do only under constraint of obligation. Kant actually leaves open the possibility that this field of agents who act in harmony with moral law but not in any way obligated by it extends beyond God; if anything other than God has a holy will, it too acts completely in conformity with reason but without any obligation.
Likewise, natural law theories of obligation, the other major family of intrinsic-authority theory, hold that some things can, as law, have authority in and of themselves, namely, the first principles of practical reason. However, the most natural explanation for why these things have intrinsic authority is that they have been promulgated as law insofar as God has made beings that participate in eternal reason; and then the person who wants to accept this view but hold that God has obligations runs into self-obligation problems roughly analogous to those we saw with divine command theory.
These are by no means the only theories of obligation (Malebranche's theory of obligation, for instance, on which God does have obligations, fits none of these very well, although I suppose it's strictly speaking an intrinsic-authority view very different from either Kantianism or natural law theory), and obviously one could do some mixing and matching (holding that, for instance, sentiment is the foundation of obligation but that certain authorities can also, as a result, obligate externally) -- and, indeed, to account for many of the things we call obligations, one would arguably have to do some mixing and matching. And it could very well be that, for instance, one could have a divine command theory with a plausible answer to the challenge I raised above with regard to divine command theories of obligation. But (1) these seem to be far and away the most popular kinds of account historically, and (2) there are reasons in each case at least to worry about the claim that God has recognizable obligations.
For myself, I tend strongly toward an intrinsic-authority theory of natural law type, and I don't think we can make much sense of divine obligations in a strict and proper sense. I find, though, that people usually raise this kind of question as a challenge for Christians (or theists of any stripe) because they are assuming that if God has no obligations then anything goes and God could just as easily be a monster as otherwise. The brief discussion of Kant above at least shows that this cannot always be assumed; and there are good reasons in general for thinking that something can be good, and known to be good, without being obligatory, or known to be obligatory. Thus the question is not really relevant to whether God is good, except on the debatable assumption that God can only be good if He acts according to divine obligations.
So I don't think this is a hugely challenging question for Christians (or theists) in general. The really challenging question here is the nature of obligation, which has for literally centuries been a very contentious and controversial debate. Given an answer to that question, one's answer to the question of divine obligations falls out immediately. But the question of the nature of obligation is a question that arises for everyone.