The lesson that only God can be a utilitarian is of even greater importance when we consider the so-called problem of evil.
As I argued in an old paper, The Problems of Evil, the problem of evil is really that of "disproportionate suffering" (DS). We don't object to people suffering as much as they deserve; some of us are even willing to concede that it wouldn't be so bad if some people ended up suffering less than they deserve. What gets many people indignant and contemptuous toward God is that there seems to be much more suffering than innocent human beings deserve, or even than some not-so-innocent human beings deserve; for that matter, any suffering they undergo is more than infants deserve, and is thus disproportionate in that respect. Understood in such general terms, DS motivates what is considered a logical challenge to classical theism. For, it is said, if God is "all-powerful" he could prevent or eliminate DS and, if God is perfectly good, he would want to eliminate it; but he doesn't, so he's either not all-powerful or not perfectly good. But as launched, the challenge doesn't quite hit the target. The questions are not whether God could or would want to prevent or eliminate DS simpliciter, but at what cost to creation he could so and whether the cost is one that his perfect goodness would impel him to pay. Thus we must consider whether there could be goods which, facilitated or occasioned by such suffering, would outweigh and thus justify the DS there actually is; if there could be, then God might not be able to prevent or eliminate such DS at a cost his goodness would impel him to pay. That begins to sound like a problem for a utilitarian.
To be sure, if there is such good, we cannot know what it is, or at least know enough of it to get the full picture; only God can and does, and so only God is in a position to make it known should, for good reason, he choose to do so. But by the same token, neither can we know that there couldn't be such a good. Hence, when the question at hand is the moral status of the universe, one must note not merely that only God can be a utilitarian; only God may be one. As I said in "The Problems of Evil," we necessarily lack the knowledge we would need to have in order to put God in the dock and convict him—just as we lack the viewpoint we would need to adopt to be successful utilitarians in our dealings with one another.