When young Frank tests his jetpack prior to the 1964 World's Fair, he wears a jersey with Adidas-trademark triple stripes on the sleeves. In 1964, Adidas only made sneakers. They didn't sell clothing until the introduction of the Franz Beckenbauer tracksuit in 1967.
Eddie Newton bails Casey out of jail a few hours after she is arrested. Casey presumably would've faced federal charges for attempting to break into a federal facility. In the United States, bail in federal cases is determined in a bail hearing by a magistrate in federal district court, and generally must be paid in cash. State criminal charges often have prescribed bail that can be paid by a bondsman soon after arrest. Despite his connections at NASA, it's highly unlikely that Eddie could've arranged a bail hearing and paid her bail in a matter of hours in the middle of the night. She more likely would've been held until the next business day to allow time for the hearing and payment arrangements. Conversely, if Eddie had convinced NASA police not to charge her yet, there would be no reason for him to tell her that he'd bailed her out.
Casey uses a cellphone to control a drone before she is arrested. She makes bail and collects everything, but the phone is missing until she later wakes her brother to get her into the Internet because her phone, which has reappeared, got soaked in the swamp.
When young Frank Walker is been taken to Tomorrowland via the It's A Small World water ride, he goes down a deep water slide which splashes a lot water over him and his bag. When he arrives in Tomorrowland, he and his bag are dry.
When Casey is outside at Frank's house in the rain looking at the satellite dishes and following the cables, her jacket goes from unzipped to zipped up and back. At one point the jacket is zipped up and soaking wet. Other times the brown jacket and the green one underneath appear dry.
When Casey feels the pin in her dad's nightstand, it takes her to Tomorrowland. She goes to the right of the nightstand, then runs into the wall, then goes to the left. When she finds the door and walks forward, she falls down the steps. When she is back in her home, the stairs aren't straight from the door, but to the left of her dad's room, and Casey never turned left out of the door.
When Casey is waiting outside Frank's house about an hour into the movie, it is raining heavily. When Frank runs out to douse the flames on the heavy machinery, the rain has stopped. Frank gets blown back by the force field moments later, but he doesn't get muddy. The rain begins again and starts falling heavily while he's lying on the ground, with no effect on his clothing. In the subsequent scenes neither his clothes or hair show any sign of rain.
At the very beginning Casey's first tour of Tomorrowland, she watches the 3 guys use the jetpacks. She is standing on a large balcony area with one almost unused, stairway visible in the distance by the Albert Einstein quote. After the 3 guys take off their jetpacks, she walks over in the direction of the racks holding the jetpacks, then looks up as a "hoverrail" train flies over her. When the scene shifts back down, a giant staircase, with many people on it, has appeared behind her. That staircase hadn't existed a few seconds before. Either that, or Casey changed the direction she was walking and ran about 100 feet in the 1 or 2 seconds the view was shifted up at the hover train.
When the Blast From the Past robots blow up and destroy the store, Casey lands in the street as a colorful bit of memorabilia slides under her. When the camera switches to the other side, and she begins to stand up, the debris under her changes.
Young Casey points out the stars on March 9, 2003m and the sky is very dark. She mentions several stars, including Sirius and Vega. Sirius is a winter star and Vega is a summer star. These stars are on opposite sides of the sky. They can only be seen together in early February, just after sunset. Vega would be setting and Sirius would be rising. Procyon, Capella, and Betelgeuse can all be seen near Sirius. Canopus is near Sirius too, but it's a southern star that can only be seen close to the horizon in North America. Arcturus is closer to Vega and is best seen in in the northern hemisphere in late spring to summer.
The girls escape in a newer-model burgundy pickup truck, with the rear window broken out. When she picks them up in the woods, the truck has is now older and dark red or black with a repaired window and grey side panels. They could have ditched the first truck and stolen the second one.
As Casey and Athena drive to Frank's house, it never seems very windy inside the truck, despite the fact that they're supposedly driving at highway speed with a gaping hole in the window immediately behind their heads.
After Casey is bailed out of jail, she holds the pin and "TomorrowLand" is shown while her father is driving. When her father grabs the pin from her after screaming "You're not seeing this!??", his hand grabs hers over the green screen.
When Casey arrives at Frank's house, the dog starts and ends as a German shepherd, but at the fullest extent of its chain it's a flatter-muzzled breed (Mastiff, perhaps?). It's a hologram, so that may indicate a programming glitch.
Casey's position in the real world seems to correlate with her position in Tomorrowland; she moves simultaneously in both worlds when touching the pin. However, it's just an illusion; the pins don't show the real Tomorrowland. It's completely plausible for people from all over the world ending up in the same field after touching their pins. They're in a kind of virtual reality, not actually in the same place.