Edgar Allan Poe does not tell us directly about the time and place of his famous story, "The Tell-Tale Heart."
If we read the story carefully, however, we can see that it most likely took place around the years that Poe lived, which were from 1809 to 1849.
Here are some hints about when the story takes place:
a) The narrator talks about using a "dark lantern" to shine a light on the old man's eye. A dark lantern is a sort of candle-holder enclosed in a dark metal box that has only a small opening to allow light to shine out (see the first link below for a picture).
b) When the narrator accidentally wakes the old man on the eighth night, the old man is scared, but he tries to dismiss the noise he heard by saying, "It is nothing but the wind in the chimney." I'm a lot older than you, but I've never been woken up by wind in the chimney!
c) When police come to investigate the situation, they say that "information had been lodged at the police office." This sounds like someone had actually gone to the police office to report his or her suspicion--not by telephone.
You are right that it is not so clear where the story takes place. Since Poe does not tell us much about this, it is safe to assume that the story takes place in the U.S.A., where Poe lived. Poe lived in a number of different cities and states: Boston, New York City, Virginia, Philadelphia, and Baltimore. I don't see any real clue as to which of these cities might be the location of "The Tell-Tale Heart."
Although the place and time are not specifically mentioned in the short story, "The Tell-Tale Heart," we do know that author Edgar Allen Poe first published the story in 1843, six years before his death. The events of the story fit appropriately within this time period, so it is likely a contemporary account taking sometime in the early to mid 19th century. Poe first published the story in the Boston Pioneer, so the story could well be set in that city, if not the northeastern United States. However, nowhere does the story mention any specific locales.