The story description that includes dramatic irony is:
"A story follows a man’s journey as he searches for his cat, but at the end readers find out the cat went missing years before and the man’s neighbors have just been kind enough to humor him each time he searches."
In this scenario, readers know something that the main character does not—specifically, that the cat has been missing for years, leading to a situation where the man's search is futile while the neighbors are aware of his misguided efforts.