I suspect the birds would eventually ignore the drones unless the drones are made to pose an actual physical threat to them. City pigeons are so accustomed to cars and humans that they barely move out of the way in time to avoid being crushed.
Chasing requires fleeing. If the bird knows from experience that the drone won't hurt him, then why would he flee from it?
[1] https://en.wikipedia.org/wiki/Migratory_Bird_Treaty_Act_of_1...
The problem there is that the bird has no time to react. Planes are way faster and actually suck air.
Come to think of it, birds don't even make enough of an effort to avoid existing aircraft that are large, loud, and do pose a threat to them, which is why these drones are being developed in the first place.
Oh boy, there goes a future evening...