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Continuity: A Bob-Omb explodes under the devolving Koopa and sends him flying. The shot shows him flying straight up, and without cutting, shows him begin to fall along the same trajectory. The big pot thing is off to his left, and well out of reach of him falling. Yet then it cuts to him falling at an obvious angle and into the pot. The trajectory was all wrong.
Continuity: Just after the car chase scene, Mario drives the police car into a tunnel, which disconnects the vehicle from the power grid. The car even says, power shutting down, causing the monitor to shut down, and for the breaks to stop working. However, even after the car leaves the tunnel and is dangling in fungus, way out of reach of the power, the headlights and the police lights of the car are still lit. They should have shut down also when power was lost.
Other: In the police car chase after the prison break, Mario lets go of the wheel to prove a point to Luigi. In doing so, their police car rams head on with a civilian car, causing the police car to ramp up and stop on top of the civilian car. Normally, this could theoretically happen. However, in that world, the police cars are fitted with a snow plow like thing on the front of the cars, with an angle to them. The way the cars hit, it is not possible for the police car to have hopped up on top of the civilian car in such a manner. If anything, the snow plow would have caused the civilian car to ramp up on the police car.
Revealing: In the prison escape scene, Mario and Luigi get on a zip line like pulley track and swing though the prison. Goombas are shooting fire balls at them. The first fire ball slowly goes and hits a wall. Watching closely, you can see the line of fire travel down the wall, showing where the special effects crew had pre-coated the wall with a flammable fluid.
Revealing: After the old lady steals the meteor rock from Luigi, the big lady in red picks her up and throws her off down below into a moving car. She gets up and shocks the driver, causing the car to rear end the vehicle in front of it. If you watch closely, you see the engine top shift forward and the back of it lifts up. This reveals it is just a fake prop attached to the vehicle and not a real engine.
Continuity: When Mario and Luigi follow Koopa's goons into the cave, Luigi jumps through the portal in the wall but Mario is more hesitant. When he falls toward it, in one shot his feet have barely left the ground, but in the next shot immediately afterwards he is pretty much upside down as he falls through the wall.