disco_air
Very simply it has no effect on the weight of the aircraft. You have to ask yourself:
1. What is supporting the birds while they are flying around the cabin of the aircraft? Answer: The air in the cabin.
2. What is supporting the air in the cabin?
Answer: The aircraft.
If the aircraft is supporting the air and the air is supporting the flying birds the net effect or difference is zero compared to if they were perched somewhere in the cabin.