Sometimes, opponents of utilitarianism make arguments that presuppose that Utilitarianism is the best school of morality, and then try to argue against it. Seems a bit silly, doesn’t it? Even so, such arguments are almost embarrassingly common. Here are some examples.
Argument 1: Utilitarianism is impractical.
One problem with utilitarianism is that it is impractical to stop to calculate the utility of the expected outcomes of our various options every time that we have to make a decision. 
The site that provides this argument also provides the refutation:
The utilitarian has an answer to this, though: if making careful calculations for every decision doesn’t maximise utility, then we ought not do so; as we’re better, in most cases, to make a rough estimate (which we generally do) and then just get on with it, that’s what utilitarianism says that we should do.
John Stuart Mill offers a different explanation in his definitive work, Utilitarianism:
Again, defenders of utility often find themselves called upon to reply to such objections as this- that there is not time, previous to action, for calculating and weighing the effects of any line of conduct on the general happiness. This is exactly as if any one were to say that it is impossible to guide our conduct by Christianity, because there is not time, on every occasion on which anything has to be done, to read through the Old and New Testaments. The answer to the objection is, that there has been ample time, namely, the whole past duration of the human species. During all that time, mankind have been learning by experience the tendencies of actions; on which experience all the prudence, as well as all the morality of life, are dependent. People talk as if the commencement of this course of experience had hitherto been put off, and as if, at the moment when some man feels tempted to meddle with the property or life of another, he had to begin considering for the first time whether murder and theft are injurious to human happiness. Even then I do not think that he would find the question very puzzling; but, at all events, the matter is now done to his hand.
Argument 2: Utilitarianism is too demanding.
Utilitarianism holds that we ought always to do whatever it is that maximises utility. That places a great burden upon us. Every time I read a newspaper, or watch TV, there’s something else that I could do (e.g. help out at the homeless shelter, write a letter to my grandmother) that will bring more utility into the world. If utilitarianism is right, then reading a newspaper is therefore morally wrong. According to utilitarianism, those of us who aren’t facing great hardship ought always to be helping those that are, because that’s what maximises utility. That, though, is implausibly demanding; reading a newspaper isn’t a sin. [1, 2]
Utilitarianism doesn’t say you’re a bad person for reading a newspaper; it only says that it would be better if you did something that more effectively increased utility. The only case in which Utilitarianism could be considered too demanding is if, when people acted in morally suboptimal ways, they were treated as if they were evil. But treating people as evil for lapses in utility-maximizing would not maximize utility.
Argument 3: It is impossible to know what will make others happy.
Each of us is intimately familiar with our own individual wants and needs. Moreover, each of us is uniquely placed to pursue those wants and needs effectively. At the same time, we know the desires and needs of others only imperfectly, and we are not well situated to pursue them. Therefore, it is reasonable to believe that if we set out to be ‘our brother’s keeper,’ we would often bungle the job and end up doing more mischief than good.
This is James Rachels’ argument for ethical egoism. It is probably the silliest argument of all, because it essentially says “maximizing utility will not maximize utility.” In trying to argue for ethical egoism it presupposes that Utilitarianism is correct. Arguing that trying to help others will end up hurting them is a fundamentally utilitarian argument. The conclusion to draw from Rachels’ argument is not that ethical egoism is superior to Utilitarianism, but that the best way to increase utility is for each person to be primarily concerned with himself.
In fact, many of the supporting arguments for ethical egoism are actually arguments for Utilitarianism.