There is no one place where Zaroff says that people do not have rights, but from what he says and how he acts, we can say that Zaroff does not believe that people have the right to control their own lives.
First, Zaroff, in a conversation with Rainsford, says that he has created a new animal to hunt. What he means by this cryptic statement is that he hunts humans. From this perspective, people have no rights, as they are hunted against their own will. Here is the quote:
The general smiled the quiet smile of one who has faced an obstacle and surmounted it with success. "I had to invent a new animal to hunt," he said.
Second, that Zaroff pressed Rainsford to be hunted shows that he is dead serious. More importantly, he did not view Rainsford's wishes as important. If you recall, Rainsford wanted to get off the island right away. This shows that he does not believe that people have inalienable rights. In many ways, he embodies, to a perverse degree, Rainsford's sentiment at the beginning of the story that there are only two classes, the hunted and the huntee.
In short, Zaroff is a demented man who does not believe that people should have the right to choose their own destiny.