I've been using a flash to photograph birds since I started about fifteen years ago. Mostly I just use the flash inside the forest, and mostly just to fill in shadows and maybe give me a slightly shorter exposure time. There are many who have prejudices against the flash, and I wanted to cover some misconceptions and also provide some tips.
In Ecuador a farmer named Angel Paz became famous when he taught antpittas and other species to come out of hiding for worms. His feat has been replicated, but at the time it was akin to a miracle. Angel never allowed use of the flash at his site, and many in Ecuador have concluded that the flash is the worst thing for birds. At Wild Sumaco lodge in Ecuador there are trails with "NO FLASH" signs. On my last visit there, one of the owners was telling what success some researchers were having using sound playback on Buckley's Forest Falcon, something I found unacceptable because of how it might disrupt the behavior of this rare bird. At REGUA in Brazil a volunteer guide who used playback all day to show birds to the guests told them that they Long-tailed Potoo might not be there because I had used a flash on the previous visit. And this is a bird 20 meters off the ground with its eyes closed. In my opinion, it should be obvious that people entering the bird's habitat, and playback are both much more disruptive than the flash. An scientific article mentioned later also concludes flash photography in the day has an insignificant effect on animals.
How do birds react to flash photography? At Rio Blanco in Colombia, Albeiro Uribe was perhaps the second person to replicate bringing antpittas out for worms. I've photographed six species of antpittas there with a flash, and not one time has one even flinched. Antbirds and birds that are 100% based on the forest floor seem to totally ignore the flash. Other species occasionally flinch or fly off when caught by surprise by the flash.
Can a flash when used at night damage the eyes of night birds? I generally keep the flash set at default or lower, and the birds are usually at least ten meters away, if not much further. The light from a flash dissipates rapidly over distance, unlike a laser. I personally have never noticed any change in behavior or difficulty flying with the birds I've photographed. One anecdotal piece of evidence is the San Isidro owls in Ecuador. They sit out in the open near the lodge and sometimes a row of photographers take photos of them for maybe even hours. Are these birds still ok? Some articles suggest there could be an accumulated effect. This article is a good starting point for this topic. The author recommends to 'err on the side of caution' and not use the flash, but that there is no concrete evidence of long term damage.
Some photographers pride themselves on not using a flash. They only use 'natural light' as if it was better or made the photos better. I prefer to not use flash when possible, but don't accept that somehow using the flash might create an 'unnatural' photo. Of course the goal with the flash is to make the photo look better than it would have if the flash hadn't been used.
Often it is said that the flash 'changes to color of the bird'. I would argue that is most cases it is the directional light from the flash that causes the color change. Some forest birds are almost never seen outside of the shadows of the forest. And there are many things that change the color of a bird. One is the exposure time. A good example of this is the Cock-of-the-rock, a highly reflective orange bird. Adding just a third of a exposure stop can make this bird go from orange to red. Shadows or lack of light obviously affect the color. A bird can have a brightly-colored belly, but if there is no light to show it, you might see some dark brownish greyish color with just a touch of the actual color. The forest itself can change the color of the bird. Light passing through green leaves will leave the photo with a green cast. Another example is evening light, which has a much larger red component; unmodified photos in this light can look very unusual.
Sometimes it is said that photos taken with a flash look unnatural or bad. In most cases I use the flash to fill in shadows and give a little life to the colors. If I had natural light directly behind my camera so as not to produce shadows, and not too strong to produce an image with too much contrast, I would of course not use the flash. But especially inside the forest, conditions are rarely what you would like. It is true that if the bird is close to the flash or the flash is set very high, the images can have a flashed look, and that should be avoided. I believe with just the right amount of 'fill flash' for the shadows, you can create a photograph of a bird in the shadows that looks as if it was taken out in the open. In the forest I use a tripod and Aperture Priority, and often photograph at very low shutter speeds, such as 1/40. A high exposure compensation setting, and a low flash setting ensures that a good percentage of the light comes from the ambient light, and that I don't get a flashed look. Some photographers use high ISO settings in place of the flash. Modern cameras handle high ISO values much better than older cameras, producing much less grainy images. But the 'fill' feature of the flash can't be reproduced.
One thing to note, using the flash is more difficult than not. With regular photography, you need to consider the shutter speed, aperture setting, and ISO setting. Adding a flash adds one more variable and many more ways you can go wrong. I've certainly lost photos by having the flash setting or the overall combination of settings incorrect.
In conclusion, I wish I could photograph every bird at 8:00 in the morning with the sun directly behind me and a thin layer of cloud to diffuse the light a bit, but that is not realistic, especially with forest birds.
This Black-headed Antbird was an unidentifiable shadow in the rainforest understory. A high exposure compensation setting and a low flash settings created a slightly dark but uniformly lit image without shadows.
This Planalto Woodcreeper was shot in near dark in manual mode with an exposure time of 1/160 (selected for a hand-held shot), and brightened greatly. The photo is acceptable because the bird is uniformly colored without any reflective colors. A tanager photographed with this technique looked very artificially lit.
This Black-billed Flycatcher was in acceptable light to photograph without a flash, and I am including both photos, first with flash. Both I think are good and which is better in my opinion is personal preference. Note the exposure time on the photos is 1/15 and 1/8. Sometimes the shorter exposure time for flash photos makes the difference. At 600mm, 1/15 is too long.