The idea of moral rights is problematic for utilitarians because this idea implies that people have rights that must be protected regardless of the impact that this might have on a larger group of people. The idea of a moral right implies that there is never any reason that justifies taking that right away.
If people have moral rights, there is never a justification for taking away those rights. When Thomas Jefferson said, in the Declaration of Independence, that our fundamental rights are inalienable, he meant that they cannot be renounced and they cannot be taken away. This sort of idea is, for example, behind our ban on torture. Even if I know something that can prevent a large number of deaths from occurring, I have a moral right not to be tortured to extract that information.
This is a problem for utilitarians because they believe that actions are good if they provide the greatest good to the greatest number of people. In such a situation, it is easy to see where torturing me (which only hurts me) will be a good thing if it preserves the lives of a large number of other people. They good that they get from living outweighs the bad that I suffer from being tortured.
Thus, the idea of moral rights presents a problem for utilitarians because it calls into question the idea that good and bad can be determined through the results of actions.