De-Kahnstructing Henry - S3-E13
Continuity mistake: When Kahn comes home after getting his promotion, Minh has the fridge door open. Between shots almost all the items in the door disappear, and the cans inside the fridge change colour.
The Wedding of Bobby Hill - S3-E14
Continuity mistake: As Bobby visits Connie and is at her window he lays his hands on the window sill. The next shot they are back by his side and they keep alternating back and forth even though he never moves them between shots. At the end of the scene Connie shuts the window smoothly even though his fingers were still there in the shot before.
Continuity mistake: In the scene where Bill is trying to open the beer can with the back hoe, he rips up the pavement with it. As Hank sees the mess he gets in the machine and pushes the pavement back that Bill tore up. When Boomhauer comes along you can see his foot falls into the hole and gets stuck even though Hank pushed it back. Also if you look the soil Bill dug up has disappeared when Boomhauer comes along.
Continuity mistake: When Bobby & Peggy are about to steal Bill's flag, the truck drives right up against the flagpole. In the next shot Bobby is standing next to the flagpole and there are a few changes. The first is that the truck has a bit of space between the back of it and the pole, whilst the first shot it had no space in between (as it hits the pole). The other change is where the rope is knotted on the pole. In the first shot it was on the opposite side of where Peggy was standing. The next shot she has not moved but the rope is right next to Peggy facing her.
The Unbearable Blindness of Laying - S2-E11
Continuity mistake: When the family leaves for the basketball game Peggy brings the wrong foam finger (not the basketball one) so Hank turns around to get it. It shows him walking into the house, finding it but also finding what he thinks is laundry all over the floor. He also hears what he thinks is innocently the washer or dryer running. He follows the noise towards the kitchen, slowly opens the door to find his mother and Gary presumably having sex on the kitchen table. The problem here is that the kitchen has no door. From inside the kitchen in any episode it is never visible. I checked and further into this episode while opening presents it shows an entrance to the kitchen from the living area where Hank entered and there is no door on that side either. In truth it is an open doorway.
Character mistake: Hank seems surprised to learn a lot of the Alamo's defenders weren't from Texas. Given that kids study this in American history, especially Davy Crockett's role in the historic event, it seems unlikely he'd be ignorant about this.
Character mistake: Hank says that his parents haven't spoken in four years. However, Cotton and Tilly were both at the Hill home in the episode "Nine Pretty Darn Angry Men" (Season 3). Cotton bullied Tilly on that occasion. And four years has not passed in between those episodes, or else Bobby would be a high school student.
Continuity mistake: In earlier episodes, Luanne is very knowledgeable about cars. This is shown in "Shins of the Father" (Season 1). In "Jumpin' Crack Bass" (Season 2), she does repairs on Hank's truck - and Hank is known to be protective of his truck. As the series goes on, she becomes a stereotypical "dumb" promiscuous type, with no apparent expertise in cars.
Born Again on the Fourth of July - S13-E14
Factual error: When Bobby and Lucky first walk into the church service that is taking place inside the karate studio, the pastor is calling everyone a sinner, and proceeds to point to a picture of Bruce Lee behind him and says he was a big sinner who repented and knew that the unrighteous would not inherit the kingdom of Heaven. Bruce Lee was an atheist.
De-Kahnstructing Henry - S3-E13
Continuity mistake: At Strickland Propane, when Kahn hands Hank his propane tank, it is set down on the counter right beside the bell. However, in the shots facing Kahn, the bell is directly in front of him, but the propane tank is nowhere to be seen. Additionally, during this scene note the difference in the ceiling height in relation to the wall boards and the doorway behind Hank. (00:02:20)
Hank and the Great Glass Elevator - S5-E11
Continuity mistake: After the banquet dinner where Bill sees Lenore, she comes to visit him at home. They are sitting down on the couch and when she's talking she has her hand on his chest and you can see there is a pen in Bill's right shirt pocket. Between shots she talks about stealing Bill away at the prom and in one quick shot the pen disappears and the next shot it returns. Also look at Lenore's hand in this scene. It goes from being next to his chest, away from his chest back to being on his chest.
Continuity mistake: The door to the slaughterhouse (which Luanne blocks with the barrel) changes configurations between shots. When Trip is trying to get in, the exterior is blue with a doorknob. After Peggy busts in, it is white with a pushbar across the middle. The interior is white with a pushbar in all shots.
Death Buys a Timeshare - S9-E3
Plot hole: During a heatwave, the Hills are dying from the heat and even look for a pool to swim in to cool off.This makes no sense, they live in a ranch house which would have central air or at worst some window air conditioners.
Hank's Got the Willies - S1-E4
Continuity mistake: In the kitchen, Hank tells Peggy he's going back to bed and asks if she's coming. She says no, and during this moment she's holding her coffee mug by the handle, but in the next shot she has her fingers through the handle, holding the mug itself. Also, the heart on the mug moves, and a red sugar container appears in her other hand. (00:03:45)
Hank's Got the Willies - S1-E4
Continuity mistake: When Hank is talking to Peggy in the kitchen about his guitar, she is pouring herself a mug of coffee. While the mug is on the counter, it is solid white, but when the angle changes, there's a big pink heart that appears on the mug. (00:03:40)
The Order of the Straight Arrow - S1-E3
Continuity mistake: When the park ranger is threatening to arrest Hank after finding them with the bird down by the small dam, the line on his hat keeps disappearing and reappearing as he talks. (00:20:20)
Other mistake: At Kahn's barbecue, everyone likes the burgers he cooked on his charcoal grill. Peggy is among the people who crowd around the grill. But in the episode "Hank and the Great Glass Elevator", Peggy and Bobby are shocked when they eat a burger that Luanne cooked with charcoal.
The Perils of Polling - S4-E25
Continuity mistake: Luanne's penguin puppet is on her right hand, then her left hand within a shot.
Serves Me Right for Giving General George S. Patton the Bathroom Key - S13-E15
Other mistake: Here, it's made into a big deal that Cotton wants to be flushed in the same bathroom Patton used. Yet in Season 4, Episode 2, "Cotton's Plot," he successfully gets a cemetery plot in a Texas military cemetery.
Answer: In addition to Brian Katcher's response: it's supposed to be a joke that Dale is always suspicious about conspiracy theories, mysteries, etc, but he is "clueless" about his own wife having an affair. Of course, as Brian wrote, many fans think that Dale is pretending to be clueless because he doesn't want Nancy to leave him.