The time period in which "The Tell Tale Heart" is set is known as the American Renaissance. This literary movement branched out into two movements: American Romanticism and its darker, alter ego known as Southern Gothic literature.
This latter literary movement is characteristic for its realism and naturalism. Gothic literature is also the first known movement to treat the themes of inner human emotion, the reality of human weakness, and humans' capacity for evil.
Therefore, "The Tell Tale Heart" is a product of the genre in American literature that first explored the darker side of humanity, and exposed the preoccupations of a very shaky society (the early 1800's) exploring subjects never used before such as human horror, delusion, and human despair.
Edgar Allan Poe never specifies the exact time period of "The Tell-Tale Heart"; it could be the 19th century (though we know a half-century has passed and that Poe wrote the story in 1843 or before) or as much as several hundred years earlier. Had the story been written in the 20th century, it would still be plausible in most respects. Because of some of the elements such as the catacombs and the palazzo, the story has a gothic feel of the old world about it. However, many of these same aspects are still in evidence today. Ancient catacombs are still in use and still no doubt double as wine cellars in some cases. The annual carnivals are still held today, and the unusual costumes of the period are still worn. Servants are still employed by the wealthy; ancient palaces are still used as residences; and the customs of Masonic participation are still present.
Modern detective methods of today might, however, alter portions of the storyline. Cadaver-seeking dogs might be able to locate Fortunato's body; video cameras might capture the two men walking from their public meeting place. Detectives might home in on Montressor as a suspect and a detailed investigation might include the search of his entire property. The absences of some of these aspects (not in use at the time of Poe's writing) help to give the story a feel of more ancient times when such a disappearance of an important man might be written off as a mere unresolved mystery.