Birds are considered as world objects and cannot be interacted with in any way. They exist only as a distant visual effect. They can be seen circling around certain areas that contain some degree of notability, such as Megaton, Old Olney and the alien crash site.
Birds remain in the air indefinitely, and will never land.
Because birds are merely a visual effect, they can fly through any surface in the Capital Wasteland. They are also classed as objects and are little more than a moving model in the sky with little/no detail of wing animation.
Birds do possess a rudimentary 3D model, although it is essentially just a bird texture bent into a bird shape. 
The black, non-hostile birds in the Mojave Wasteland avoid direct interaction, flying away if the Courier gets too close. They are seen sparingly in the Mojave, but are relatively common in Nipton and Boulder City.
When shot at, birds will explode into several bloody pieces, but the corpses cannot be interacted with and will disappear quickly.
The Institute utilizes synth-UAVs, called "Watchers," to monitor activity within the Commonwealth. Ronnie Shaw comments on "seeing a lot of those Institute crow-things sniffing around." Justin Ayo makes numerous references to watchers, and the Synth Retention Bureau's surveillance monitor banks display footage at elevated angles or "birds-eye" view. In The Art of Fallout 4, numerous bird and bat concept arts are referred to as "Watcher Sparrow," "Watcher Crow," "Watcher Radgull," etc.
Mutated chickens can be found on the Island, and have lost almost all of their plumage due to high radiation levels and the radioactive Fog. They are docile and will flee from the player character if approached or shot at, but they can be killed and looted for chicken thighs.