Assume that you are holding a cage containing a bird. Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage?
Does it make a difference whether the cage is completely closed or it has rods to let air pass?
Answer
If the cage is completely closed, it doesn't make a difference if the bird is hovering inside it or if it sits on the ground. When flying, the bird pushes air to the ground which will exert a downward force on the cage exactly equal to the weight of the bird. This is a direct consequence of the conservation of momentum and Newton's second & third law. Since no additional external force is acting on the cage-bird-system when the bird is flying as compared to when it's not, the acceleration on the cage can be no different. The effect due to the flying bird only concerns the internal forces and since action=reaction, they cancel.
However, if the cage would not be closed, some of the 'wind' due to the bird could escape the cage and would become an external force, making the cage-bird-system lighter.
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