To burn CO2 is photosynthesis for plants, in which trees make food and energy, and in the same time trees also works on respiration on the other way to produce CO2, which consumes food and energy, but in much less amount than photosynthesis. And during nights when there is no lights, the process to burn CO2 is closed, but the process to produce CO2 is still on. You should know about this from 5th grade class!
Why are birds not affected? You should ask why humans are not affected by CO2 produced by plants in the same room. Air is everywhere, and the concentration of CO2 is much thinner than that of O2. If the bird nest ABSOLUTELY isolates from outside but only connected to the tree, then the bird would definitely die of suffocation. Actually, birds and humans are all affected by CO2 produced by tress in sleeping nights, but the effect is small. That's why too many plants in living rooms is not good.
The rate of production would be small and the CO2 highly diluted by the oxygen content of the ambient air. Air movement would aid the mixing. So birds wouldn't be affected. Also carbon dioxide is more dense than air and sinks to the ground.