A plane does weigh more when a bird is inside, as the bird exerts a downward force equal to its weight whether it is flying or sitting. When the bird flaps its wings, it temporarily may not push air downward, leading to a momentary reduction in weight. The discussion also compares this to people in a plane lifting their legs, highlighting that lifting does not exert a force on the plane, while jumping does. Newton's third law is referenced to explain the forces involved in both scenarios. Overall, while the weight variations are minimal, they do exist in both cases.