What are some ways the utilitarian theory might give the wrong result and make wrong actions right or right actions wrong?
To a utilitarian, there could be no such examples. Utilitarians say that the rightness or wrongness of an action can be seen from its results. Therefore, any action that increases the general happiness is right, regardless of the motive for which it was taken. With such a consequentialist view of right and wrong, there can be no mistakes.
Presumably, then, we must look for an example of how something that was done for a bad reason had a good result from a utilitarian point of view. We can see this in small-scale examples from day-to-day life. For example, let us say that a parent signs a child up for dance lessons because the parent simply does not want to take care of the child at home. This is presumably a bad reason to sign your child up for lessons. But what if the child then comes to love dance and it becomes a huge part of his or her life? The parent’s action was taken for a bad reason, but it ends up turning out well. It is harder to talk about such things in politics and other such “big picture” ways because the motives people have and the impacts of their actions are not always clear. However, we can at least see how this is possible. Let us imagine, for example, that the Republicans oppose “Obamacare” only because they think it is a good political move. This is a selfish reason. But then let us imagine that defeating “Obamacare” ended up being positively good for the country. This would be an example of an act that would have a bad motive, but which would be seen as a good action from a utilitarian perspective.