In all the forest domes, entry to the forest is shown to be through a tunnel in the rim of the dome. However, no connection of this kind can be seen in any of the exterior shots of the domes. Moreover, when the domes are jettisoned, the shots of the pylons connecting the domes to the ship reveal barely enough room for the ejection system engines. There does not seem to be any kind of connection which would indicate an access tunnel.
When Lowell is playing pool, the robot is dropping the balls into the center pocket. There are two small air lines connected to the ball gripper. The camera cuts to a wider shot and you can see the robot has set the ball gripper on the table. The airlines have somehow disconnected themselves. It picks up the nine ball rack with its main gripper and drops it off on the table. It then picks up the ball gripper. There are no air lines connected to the ball gripper at this moment. The movie briefly cuts back to Lowell whilst the robot retrieves the first ball. When it returns, you can clearly see the air lines have been reconnected to the ball gripper.
When Lowell is driving the vehicle into the storage with the camera on the car, a cargo container on the left hand side is seen with the top lid open, but when the camera position changes, that same container is now closed.
When repairing Huey, Freeman used hemostats to clamp on and remove a circuit board from its socket. That has never been a valid way of removing a circuit board, especially back when the movie was made. Circuit boards in sockets like that have always had special methods of being removed to prevent any shorting out or arcing of circuits, such as a puller that attaches to special holes, or spring-loaded clips on the edges that release the board when the clips are released. You never touch the surface of a board with anything metallic, even if the board is coated with an insulator.
During the space walk when Huey and Dewey show Louie's foot to Freeman, Freeman is wearing a blue space suit that looks like an ocean diver's dry suit, with a thin helmet. The suit does not look overly tight, nor is it inflated, it merely fits snugly without being skin tight. However, in space, he would need an atmosphere inside the suit, meaning it would have to be inflated, and the vacuum of space would explode that foam suit in an instant.
When Huey hands Freeman a small gray book while he is looking in a microscope. Overlooked is that Freeman has moved -- and in a later scene Freeman is in a different chair, reading a different much larger book.
The goof items below may give away important plot points.
Lowell (Bruce Dern) was arguing with another crew member about who would be best qualified to lead the assumed new forest service, citing his 8 years aboard the Valley Forge, yet he didn't realize one of the fundamental necessities for plant life later in the film when his forest was dying: light. The position of the domes, as well as the ships, must have been dictated by the need for sunlight as it would have been one of the main functions. At the very least automated systems should have warned him about losing the necessary light.
When Lowell plans to blow up the ship at the end of the movie (at around 1h 26 mins)), he is sitting with the armed nuclear charges at his feet and the detonator in his hand. As the camera pulls back, his location on the ship is obvious as the only brightly lit window on the ship, but the explosion comes from a visibly different part of the ship.