They are: of knowledge, love, and power. The problem of knowledge has already been dealt with:
Act-utilitarianism suffers from the crushing objection that "total happiness" is spread over billions of people for a million years in the future. What do I know of such things?
The problem of love is as follows. In order to bring about the greatest good for the greatest number, I have to will or desire that good. But that good, though maximized overall, is imparted into individuals. I thereby will good to those individuals, which is the definition of love. Thus, utilitarianism requires me to love people; moreover not any specific person by mankind as a whole. What kind of love is that?
It is clear that even the most outgoing person will have only a few "dear friends" whom he loves with a full-bodied love of friendship. Everyone else is a stranger to him, capable of eliciting only general "disinterested benevolence." Again, what is the nature of this love? Consider Mises' understanding of the proper emotions of the economist: Subjectivism, he says
takes the ultimate ends chosen by acting man as data, it is entirely neutral with regard to them, and it refrains from passing any value judgments. ...
If Eudaemonism says happiness, if Utilitarianism and economics say utility, we must interpret these terms in a subjectivistic way as that which acting man aims at because it is desirable in his eyes. ...
At the same time it is in this subjectivism that the objectivity of our science lies. Because it is subjectivistic and takes the value judgments of acting man as ultimate data not open to any further critical examination, it is itself above all strife of parties and factions, it is indifferent to the conflicts of all schools of dogmatism and ethical doctrines, it is free from valuations and preconceived ideas and judgments, it is universally valid and absolutely and plainly human. (HA, 21-2)
In short, an economist and now any utilitarian in regard to an arbitrary stranger proclaim:
I will to you those goods that you will to yourself. Whatever it is you want, perhaps as long as it's not criminal or especially vicious, I also desire for you, and I even root for your success from a distance.
But when interpreted so broadly and innocuously, utilitarian love ceases to have any action-guiding clout or imperative. It devolves into "I enjoy watching people strive and seek their happiness; I cheer when they find it, and grieve when they, sometimes tragically, fail; but that is all part of the work and way of the world. For each good desired by a person, call him Smith, there is already someone, namely Smith, who is pursuing it single-mindedly. I have nothing to add to this; the greatest good is already being promoted without my assistance. The world works; all is well with it; I am content; though, like all others, I, too, seek my own happiness." If one is content, where is the motivation to thrust himself into action to start maximizing overall good?
Suppose now that I were for some reason motivated to promote greatest overall good. What exactly am I supposed to do according to utilitarianism? I mean, do I help people? To do their jobs, say? Do I approach a random janitor cleaning up in a corporate building after hours and say, "Hey dude, I want to help you vacuum the floor. I don't actually care about you, but helping you will promote the greatest happiness for the greatest number, and I am commanded to do this." Isn't this more than a little absurd?
More plausibly, I might need to do Catholic works of mercy: feed the hungry, visit the sick, instruct the ignorant, admonish sinners, etc. If utilitarianism demands that these be done unto utter exhaustion, then it is an awfully ambitious doctrine. No Catholic saint probably measured up to an ideal that rigorous. The fact then that there are no actual utilitarians in the world should give pause to those who advocate utilitarianism.
Further, regarding "helping," I cannot forsake my own life and become a tool of others who will use me, quite selfishly, to pursue their own aims. I am a man, not a robot or slave whose purpose it is to serve its owners. I cannot disappear as a person and turn myself into an appendage of other people. Imagine if everybody was this sort of a helper; then everyone would be an instrument for others; and no one would have a life or his own goals or interests. There would be no one whose goals could be furthered by help from others, because everyone would be just a helper to others. Utilitarianism seems self-refuting.
This is the problem of power. We can see that all 3 problems are rather severe and undermine classical utilitarianism. As I have suggested, utilitarianism rightly understood is defensible, however.