I've been reading all the contributions to this discussion and I think everything hinges on what we mean by a closed system. I was thinking initially that contributors to this post were calling a cage a closed system as long as the birds stayed in the cage.
In physics a closed system is defined as one which does not exchange matter with its surroundings, but can exchange energy with its surroundings. An isolated system is one which does not exchange matter and/or energy with its surroundings Now a cage with birds in it is by these definitions neither a closed system nor an isolated system since it can exchange matter with its surroundings in the form of air, dust etc and can also exchange energy in the form of heat and so on.
It is clear that in nature it is impossible to have a truly closed system or an isolated system.
If, as some people have said, the flapping of the birds' wings keeps them in the air and that there is an equal and opposite reaction on the bottom of the cage then for this to work the cage would have to be a sealed unit so that no air could escape (as it could with a normal cage). But the cage must also be an isolated system as well so that the sound waves caused by the birds flapping their wings would not cause the walls of the cage to vibrate, which would be an exchange of energy to the surroundings.
So to sum up I now think that I agree that if the system is closed and isolated according to the definitions above then when the birds fly the weight of the closed system will not change.
But of course this is very difficult to arrange in practise.
Gen2: But you are bringing in the Helium from outside so that's cheating. You would have to add the weight of the Helium and so the total weight would be the same.