Poe creates tension using a number of techniques. One is the rhythm of the dialogue. The narrator says, "‘True!—nervous—very, very nervous I had been and am; but why will you say that I am mad?’’ The very pattern of interrupting himself creates tension from the first line.
Second, he does so by setting up specific claims and expectations. Why is the narrator nervous—and why would the listener think he's crazy? Unanswered questions like these create tension.
Third, the story's tight focus on the old man creates tension, as does a fourth point: the search by the authorities. There's a lot of threat there.
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The narrator included a lot of repetition that built up tension. After reading the first sentence of the story, the reader would want more to continue and understand well the story.