What do Pigeons look like using thermal?

Thermal is easy, depending what you select in the setup, you are dealing with a profile like "white hot" lighter is then hotter objects. so any mammal / coyote / bird etc will look whiter / lighter in the image, trees / foliage are typically cool and will be dark.

the bird on a wire will be very white against a black-cool sky.

*its very rare you get conditions where ground and or air temperature exceed body temp. but it is conceivable, I have a couple times experienced ground-out when california desert ground were 98deg ish and humans were invisible on the ground basically. but most of the time, specially at night, anything breathing will be hotter than the air. = white profile. (feathers don't show just the naked heating body shows)

*** BTW another thing about thermal, air is an insulator, so long range thermal is kinda impossible as the heat signature become less and less the further the distance.. you should see effective recognition out to maybe a couple hundred meters.. but not miles.
There are exceptions to that, I have seen military equipment demonstrated that product HD thermal images over miles, but this is not true for equipment most of us can buy with disposable income.. so to speak.

Hope this helps, I dont have a thermal scope but with with thermal in general.

Couple thermal images, dog is white-hot essentially, the person is also white-hot. but also shows you the range from the fridge to the surroundings and how the person stands out warmer than surroundings.
Im sure somebody have a thermal scope and can post some hunting snaps.
Basically real thermal cameras starts around 3000-4000 and up, lower than that is 180x60px resolution like in FLIR ONE products and the gazillions you can buy for smartphones. 180x60 is not useful for scope aiming. which is why you mostly see IR enabled cameras in night scopes using a illuminator rather than being actual thermal.

I would imagine a pigeon has the same thermal signature of any other bird, so here is a short video of flushing what may be quail or grouse with a thermal cam.

https://youtu.be/bpNWD3bQwDE

I have looked at ducks with my thermal at 20 yards, and they were barely visible even at 30*F, probably due to their excellent insulation.

Not a pigeon but that's my bird sitting on his heated perch. It's kind of unfocused because it was closer than the 5 meter minimum focusing distance. Very easy to spot as he, his perch, and an AC adapter are significantly warmer than the rest of the room.

As these guys said, more you spend, better the picture.

You asked about Pigeons only. Is this so you don't shoot the wrong bird at night? Just wondering.

The Military is always 30-40 years ahead of even thinking about releasing whatever they are using for civilian usage. Military grade FLIR, thermal imaging, at THEIR level allows them to see a clear enough outline THROUGH walls in a house to tell who the hostage, and who the taker is. Clear enough that if the call is made to green light or possibly loose the hostage right then, chances are not good at all for the hostage taker. Walls are no problem for monolithics as long as there is a background that will eventually stop it. Walls aren't even a problem for police any longer, but the urban setting will always bring problems & compromises.

Birds are pretty easy to spot using thermal in normal conditions. Their body temperatures are higher than many mammals at 105-120F.