The answer depends a bit on the specifics surrounding the object. (By the way: 'Weight' is objects 'mass' under the influence of gravity. For example, you might weigh a different amount if you were on the moon (lower gravity) but your mass would be the same. For this answer, I will mostly use the objects 'mass'. )
If the object is in free-fall, (ie, its thrown or falling) then the mass of the object does not effect its speed- or more specifically, the increased inertia [resistance to change] of the object caused by its increased mass is perfectly balanced by the increased force gravity on this mass. You can try this yourself with something of similar size, say two plastic bottles, one full of water, the other empty. Drop them at the same time and see which one hits the floor first.
If the object is not in free-fall, but say on a flat surface, then the force required to bring object up to a certain speed is proportional to the objects mass. For example if a car is twice as heavy (mass) as another, it requires twice as much force to bring it to the same speed (assuming you accelerate them at the same rate). There are other factors at play when an object is on a surface, such as friction, which also increase with mass and effect the objects 'speed'.