External conflict is a struggle between a character and some external or outside force, such as another character, nature, or society. In "The Tell-Tale Heart," the reader can discern two notable external conflicts: 1) between the narrator and the old man, and 2) between the narrator and the police.
The unreliable persona of the narrator discerns the old man and his creepy eye to be a threat and trouble to his very existence. Ultimately, the narrator concludes that he must murder his roommate to relieve himself of the troubling eye. Even though this interaction between narrator and the old man is extremely one-sided, the conflict culminates in a life and death struggle:
"The old man's hour had come! With a loud yell, I threw open the lantern and leaped into the room. He shrieked once—once only. In an instant I dragged him to the floor, and pulled the heavy bed over him."
This murder scene in "The Tell-Tale Heart" is an example of external conflict, man versus man.
Later, as the narrator attempts to elude the police, the reader sees an example of man versus society:
"A shriek had been heard by a neighbour during the night; suspicion of foul play had been aroused; information had been lodged at the police office, and they (the officers) had been deputed to search the premises."
The officers represent society and justice; they have a sworn duty to ensure that citizens uphold the laws of their town. The narrator's act of murder and his attempt to conceal the crime challenges that society's moral and legal codes by basically trying to outsmart the 'system.'