Why obey God?
One plausible answer is that our relationship to him entails that we have an obligation to obey him. This is the rationale offered for parental authority: the nature of the parent-child relationship entails that the child is obligated to obey the parent (generally speaking, of course). But what explains this obligation? It depends on who you're asking. Perhaps the obligation is simply a natural feature of the relationship; to ask why the obligation exists is like asking why the color yellow has the shade that it does.
This explanation appears to be the one C. Stephen Evans offers in God & Moral Obligation. He writes:
"If God exists and is a genuine person, then the relation between creature and creator is a genuine social relation, and like other relations, carries with it distinctive obligations...A proper social relation with God is one that requires [or obligates?] humans to recognize the enormous debt of gratitude they owe to God, as well as the value of an on-going relation to God. Most religious believes have seen this relation to God as one in which God rightly has authority over them" (29).
The chain of reason seems identical to the one that justifies parental authority: God and man have a social relationship, and that relationship entails a recognition of gratitude to God. This recognition generates good reasons to obey God's commands, and hence, God has authority over man.
Why does our relationship with God entail this obligation? It is not clear to me why Evan thinks so. One hint is that gratitude explains why we might be motivated to obey God. Elsewhere, he writes:
"I should want to satisfy the requirements of a being to whom I owe an unlimited debt of gratitude, and whose love for me is such that he intends me to enjoy an eternal happiness in communion with himself and others who love him" (31). In a footnote, he says that this passage indicates that he "is committed to the existence of normative facts that do not depend on God's commands..." (31).
I take this to mean the following: when someone asks why they should obey God, they should recognize their debt of gratitude to him. Upon this recognition, they recognize the normative fact of gratitude: recipients of gifts are obligated to express thanks to their gift-givers. This fact exists independently of God; it does not exist because God willed it into existence. And when we couple the recognition of our debt to God with the recognition of the gratitude principle, we find that our debt of gratitude requires of us an expression of our thanks to him. I take it that Evans thinks no less than our full obedience to God is a sufficient expression of our thanks.
There are at least three ways to object to this argument: first, we might object to the content of the principle of gratitude. We might concede that this principle of gratitude exists, but that it does not require full obedience of us. It might entail something less demanding: a thank-you card, a good reason to submit to God's authority, etc. But this objection might say that whatever gratitude requires of us, it is not full obedience to God.
The second way of objection is this: the principle of gratitude does not exist independently of God. If you're someone who thinks that God is the creator of everything, then Evans' appeal to a normative fact that does not depend on God is straightforwardly unacceptable since it is a thing God did not create. Most classical theists, I think, would surely get off board here.
The third way of objection is this: the principle of gratitude might exist, but it seems unlikely because of how strange it is. Christian moral realists often say the following to atheist moral realists: sure, it's possible moral values may somehow exist like numbers and shapes do, but don't you find that strange? How did they get here? Are they arbitrary? What authority do they have over us? Etc. But we might say the same thing about this principle: sure, it might exist, but how did it get here? Why does it have authority over us? Why is its authority so strong that it entails absolute obedience to God?
I think the second route is most persuasive for classical theists; I think the third route is most persuasive for non-classical theists. First route has some merit, too, but I don't have space to elaborate.