Typically, even with a low/zero-VOC paint base, the tint/colorant mixed in will add VOCs - generally the darker the color, the more colorant that is needed. But this also varies by company... Benjamin Moore has a paint out that is supposed to bind the colorant so that it still has zero VOCs after tinted, and Sherwin-Williams Harmony is supposed to be so low odor that you can use it in an occupied room.
The cheaper paint I've used from Lowes before, though 'low VOC', still had a strong smell and I waited 2 weeks with that before moving the birds into the room. So I think it's impossible to give a set answer, because of the variance in even low/zero VOC paints when it comes to odor. I do not and cannot move my birds into another house when painting. If I'm painting in another room, I put a towel under their door, turn their air filter up to high, and use fans to direct the fumes away from their part of the house.
Use your nose, paint when it's nice enough to open the windows to help with airing the room out, remember to look out for sense-fatigue when you're evaluating the remaining odor, and add on some extra time to accommodate birds' more sensitive respiratory systems. I'm going to use Sherwin-Williams in my dining room soon, so I will be curious to see how different it is from the el cheapo paint I've used before.
