Interesting question. They probably will not like it. It may interfere with them laying eggs, maybe that day but more likely the day after. They release the yolk to start an egg about 25 hours before the egg is actually laid so that's probably where you'd see the effect if you see one with the eggs.
I don't know what your coop and run look like, especially in relation to that tree. Or how big that small tree is or how much cutting you'll have to do to safely remove it. I'd expect them to go as far away as they can get and probably be excited. But I'd also expect them to calm down pretty quickly. Something like an initial panic then more like "I don't like it but it hasn't eaten us yet". In any case there will be no permanent harm to them. They will get over it.
The first time I mow around the run in spring the chickens freak out. They run into the coop and get all excited. But pretty soon they realize when I pass by the mower is throwing bits of green stuff into the run through the fence. By the second or third time I mow they do not run from the sound of the mower. They come running for their treats.
My chickens are used to seeing me carrying a bucket of feed or water. Doesn't scare them at all. But if I carry something dangerous, like a camera, they tend to get really nervous. It scares them because it is something different. It's like they are scared by a potential predator, maybe like a hawk or larger songbird flying overhead. They are scared for a bit but soon get over it.
If it makes you feel better toss them some treats while you are removing that tree. That's for your benefit. They'll like the treats as they always do but will be fine without them. Plan how you are going to remove that tree and take the time to do it safely. Don't injure yourself or cause property damage worrying about how scared the chickens are. They will get over it.
