An increase in a car's speed does not change its weight in terms of rest mass, which remains constant regardless of velocity. However, some argue that relativistic mass increases with speed, suggesting a connection between speed and perceived weight. The discussion also touches on the combustion of gasoline, which can affect the car's overall mass as fuel is consumed. The submarine paradox is mentioned as a potential point of confusion regarding mass and buoyancy. Overall, the relationship between speed and weight is complex and involves considerations of both physics and the specifics of the vehicle's operation.