1. The doctor told the patient that there was no medicine he could give her for her illness. He told her that she would just have to let the fever run its course.
2. There wasn’t much the police could do to stop the demonstration. Since it was peaceful, they let the demonstration run its course.

The expression is most often used in terms of illness (opt.1), but can also be used to describe other processes that have a predictable or established course of events.