As our police become more overworked, they are less likely to respond to situations where no crime has been committed than would once have been the case. However, there are still a variety of situations that can require a police response even though no crime has been committed.
Perhaps the most common situation like this is a traffic accident. In most traffic accidents, no crime has been committed. At the most, there has been some sort of violation, but sometimes, there can be a collision and no tickets will be issued. Even so, a police response to a traffic accident is expected.
Another situation is one in which a crime might have been committed, but where no crime actually has been. An example of this would be a suicide. Whenever a body is found where death was unattended, the police need to respond. They have to treat the scene as if it might be a crime until they can prove otherwise. In the case of a suicide, they have to be sure that the person really did kill themselves and that another person did not kill the victim and then arrange the scene to make it look like a suicide.
These are two potential situations in which no crime has been committed but a police response is expected.