Either God does or does not have moral obligations.
If he has moral obligations, divine command theory seems to be false. Divine command theory comes in two versions: command theory and will theory. On command theory, an action is obligatory if and only if God commands it to one. But no one can impose obligations on himself by commands (one can impose obligations on oneself by promises, of course). On will theory, an action is obligatory if and only if God wills (in a relevant sense) one to do it. But what one wills oneself to do does not impose an obligation. That's all I'll say about this horn, though more can probably be said.
If God has no moral obligations, however, then in particular he has no moral obligation to keep his promises and reveal only truths to us. But the Western monotheistic religions are founded on an utter reliance on God's promises and revelation. Without God having moral obligations, why think that God's promises and revelation are trustworthy? (It would obviously be circular to think so on the basis of God's promises and revelations.) So if God has no moral obligations, Western monotheistic religions are in trouble. But most divine command theorists accept one of the Western monotheistic religions.
Perhaps, though, it is impossible for God to break promises or lie, even though he is under no obligation to keep promises or refrain from lying. But if it is not wrong for him to do these things, why can't he do it? If it's just a brute limitation in what he can do, then that seems to conflict with his omnipotence. Maybe, though, God's inability to promise or lie follows from some other essential attribute of God.
Perhaps his goodness? But goodness in a context where duty is not at issue, i.e., deontologically unconstrained goodness, does not seem sufficient to rule out breaking promises or lying.
Maybe in the case of an omnipotent being, though, it does. Goodness is opposed to inducing false beliefs in others, since false beliefs are intrinsically bad. So in our case, deontologically unconstrained goodness might lead one to break a promise, because one made the promise in ignorance of some aspect of the consequences of keeping it, and to lie because there is no other way of achieving some good. But an omnipotent and omniscient being is not going to suffer from such limitations. Sometimes the only humanly possible way to save someone's feelings from being hurt is by lying to him, and deontologically unconstrained goodness may lead one to do that. But God can directly will to have someone's feelings not be hurt.
But this line of thought is a dangerous one to the theist. For it is pretty much the same line of thought that leads the atheist to conclude that God, if he existed, would prevent various horrendous evils. In response to the atheist, the theist has to insist that there may very well be goods—perhaps but not necessarily beyond our ken—that are served by not preventing the horrendous evils. But if we are impressed by this line of thought, we will likewise be unimpressed by the thought that whatever end might be accomplished by lying or breaking of promises can be accomplished by an omniptoent and omniscient being without these. In particular, a sceptical theist cannot give the response I gave in the preceding paragraph.
There is a different line of thought, though, that might work better, inspired by Steve Evans' version of divine command theory. In addition to the distinction between permissible and impermissible actions, there is a distinction between virtuous and vicious actions, and it is only the permissible/impermissible distinction that is grounded by divine command theory. God, one can say, is essentially virtuous. But lying and breaking promises is vicious. Hence God can't do these actions, not because they are wrong, but because they are vicious. I think this is the best response to the dilemma, but I am not convinced.
One reason I am not convinced is this line of thought. Suppose that what makes lying and promise-breaking vicious is that these things are wrong. This is actually plausible. Consider this line of thought. A lot of people think that in extreme circumstances it is permissible to lie or break a promise (we might, though, argue that an omnipotent being doesn't end up in such extreme circumstances—this may be a subtly different line of argument from one that I argued against above, I think). They aren't going to say that lying or breaking promises is always vicious—only that it is vicious when it is wrong, and then because it is wrong. A minority of people, including me, think lying is always wrong (I don't know the promise literature, so I won't talk about promises here). They presumably think lying is always vicious. But surely it is always vicious precisely because it is always wrong. If so, then it is quite plausible that lying and promise-breaking are vicious because, and to the extent that, they are wrong. But the divine command theorist who says that they're vicious but not wrong for God cannot take this line.
Another plausible view is that lying and promise-breaking are wrong, when they are wrong, because they are vicious. But again a divine command theorist cannot take this line of thought, because that would allow one to ground wrongness facts in non-deontological virtue fact, and would make divine command theory unnecessary.
What the divine command theorist needs to hold here is that there is no explanatory relationship between the wrongness of lying and promise breaking and the viciousness of these. And that doesn't seem very plausible, though I do not have a knock-down argument against that.