After nearly drowning in the tub (due to the weight of a wet sponge upon his head), Mr. Burns realizes that if he had died, there would be no one to look over his power plant, so he decides to hold tryouts for local kids to be his heir. Mr. Burns rejects Bart in a special way, by using a machine to kick him in the butt. He gets revenge on him by smashing his windows, breaking his statues, and filling his car with water. Mr. Burns notices his evilness and makes him his heir.
Homer eagerly accepts, though Marge feels that since Bart will be inheriting Mr. Burns' money, he should spend some time with him. Bart does and initially hates it. Mr. Burns tries to get a pizza delivered by Krusty the Klown to cheer Bart up. However he clicks the wrong button and it ties Bart down to his chair and gags him with metal bars. After setting him free he clicks the correct button and Krusty delivers his pizza. Bart then begins to love living with Mr. Burns as he gets his way all the time. Once he returns to his family's lifestyle, Bart grows upset that he can't get his way all the time, and returns to Mr. Burns. The family tries to get Bart back, but both the Police and attorney Lionel Hutz prove to be no use. Homer's next attempt to get a P.I. to help results goes awry. The P.I. brainwashes Hans Moleman into thinking he's the Simpsons' son, Bart. However, the P.I.'s attempt doesn't fool Marge and she soon fires him for not properly doing his job.
Bart starts to feel lonely from the lack of fun, and wants to go back to his family. However, Mr. Burns shows Bart a video revealing that his family no longer cares about him—even though the Simpson family members in the recording are obviously actors hired by Mr. Burns to portray negative versions of them. Bart slowly grows to like Mr. Burns even more, and is invited to the Power Plant to fire employees. Bart finds this little game fun, until he is told to fire Homer. Bart refuses, and instead attempts to fire Mr. Burns. This backlash causes Burns to claim that Bart can never be his heir or son.
Bart is then reunited with the Simpson family, who explain that they knew about the actors as they hung out with them to get their characters down. They needed to trick Mr. Burns into thinking they don't need Bart anymore. Marge makes the point that the real Simpson family members truly missed Bart, and Homer introduces Bart to his new brother Hans Moleman, whom Marge wants out of the house. Homer is aghast, but eventually agrees to be rid of Hans.
In "The Simpsons 138th Episode Spectacular", a deleted scene that was supposed to be in "Burns' Heir" was shown. When Homer turns up to Burns Manor asking for Bart, Mr. Burns deploys a robotic Richard Simmons which begins playing "(Shake, Shake, Shake) Shake Your Booty". Homer runs away, and not long after that, the robot begins to go haywire. Waylon Smithers tries to shoot it but to no effect, and Mr. Burns, Smithers and Bart take cover in the mansion before it explodes.
The reason this scene never aired was because during test screenings, the show's writers did not find the scene funny.