The title is a play on words, taken from the name of a type of German meatloaf, Falscher Hase (in English, "false hare").
The Big Bad Wolf—worshiped throughout the cartoon by his nephew, who calls him "Uncle Big Bad"—invites Bugs Bunny to join the Club Del Conejo, a club for rabbits. Bugs knows that is a trap but plays along, as things had been getting kind of dull lately.
Big Bad admits Bugs in, and tricks him into signing an insurance form. Bugs' first initiation test is to ring a bell, rigged to cut a rope and let a safe fall on him. Bugs ticks off Big Bad by ringing the bell with a nail and a nickel. When Big Bad tells Bugs to hit the bell, Bugs merely flicks it, causing Big Bad to come up and demonstrate himself. Big Bad, however, gets flattened in the process.
Big Bad then tests out his next plan, which involves shouting "now" to his nephew, so Big Bad's nephew will fling open a closet door, rigged to close a spiked casket on Bugs. Big Bad then tells Bugs they're ready for his club picture. Bugs pulls all sorts of pose, making Big Bad come up and demonstrate by showing him it's "straight and arms down." Bugs immediately says "I get it now," which makes the nephew close the casket on his uncle. Bugs then asks the nephew to let him know what develops when Big Bad comes out of the darkroom. The nephew then peeks into the casket, and then closes it again, cringing.
Later, Big Bad tells Bugs to crawl through a hole (actually, a fake wall, with a cannon in the hole) and await further instructions. While Big Bad tells his nephew to pull the cord when he gives him the signal, Bugs paints another hole in the wall. When Big Bad sees there are two holes, Bugs tricks him into going into the booby-trapped one, so Big Bad gets blasted through the wall. Bugs then flips the wall so that when Big Bad demands they try again, he prevents Bugs from going into the booby-trapped one and gets blasted through the wall again.
Finally, Big Bad directs Bugs to climb inside a hollowed-out tree and "await further instructions." Bugs climbs out when he sees the wolf pack it with explosives. The dynamite detonates and causes the tree to crash onto the Club Del Conejo. The little wolf asks what to do next, and a clobbered Big Bad suggests opening a chicken club, and Foghorn Leghorn immediately appears saying "Somebo--I say, somebody mentioned my name?" The cartoon ends with Bugs and Big Bad's nephew wondering if Big Bad or Foghorn can "chicken out first."
This cartoon was the last production completed by the original Warner Bros. Cartoons studio, and is also the last to feature the "bullseye" opening and closing title cards, and the long-familiar version of "The Merry-Go-Round Broke Down" theme as well as the usual "That's all Folks!" sendoffs. This marked the final appearance of Bugs Bunny and Foghorn Leghorn during the Golden Age of American Animation, as they would not appear in new cartoons and media for the next 16 years. False Hare was the second-to-last short from the original studio to be released.
When this cartoon aired on CBS in the 1970s and 1980s, the scene of the Big Bad Wolf making Bugs pose in front of an iron maiden was cut to remove the Big Bad Wolf showing Bugs how to pose in front of it, Bugs saying, "Oh, I get it now!", and the aftermath, which implies that the Wolf was impaled.