I might as well give my opinion. Take it for what you think it is worth.
I think your question,
What can I NOT put in the compost pile? has two elements. One, what are you going to use the compost for? If it is for vegetables, you do not want to put in dog or cat poop, diseased or insect infested plants, or obnoxious weeds with seeds. An open compost pile that chickens can get to is not going to cook enough to kill all pathogens, insects, disease spores or seeds. Even if your chickens do not have access to an open pile and no matter how much you turn it, it will not cook every thing perfectly. There will always be some areas that do not get cooked. It is the temperature that kills a lot of these things, not the rotting process. If you are using it for something non-edible, then it does not matter as much, but make sure you wash your hands.
The second is what should chickens not eat? I'll attach a Treats Chart that shows what to feed chickens. Near the bottom is a list of what not to feed chickens. I don't 100% agree with the chart but it is much better than anything I have. You will have some people look at the chart and say, I feed that stuff to my chickens all the time and it doesn't hurt them. First, you really need to read he chart. It does not say don't feed potatoes or potato peelings to chickens. It says don't feed the green potato peelings to chickens. Solaline is the chemical in green potato peelings that is bad for chickens and humans. (I looked it up.) It does not say don't feed beans to chickens. It says do not feed uncooked or undercooked beans to chickens. There is a difference.
It is like Punkinpeep said, it is a matter of moderation. It is not that the chicken is going to fall over dead the instant it eats an uncooked bean. Small amounts of many things won't hurt the chickens. They will process the poisons and get them out of their systems before they reach concentrations high enough to cause problems. Some things will build up in the system though and either not be processed out of the system or be processed so slow that they can still build up. I don't know what these things are or the rate they are processed. Some of these things don't kill the chicken. They cause damage other ways. The kidneys and liver are two of the organs at the greatest risk since they have to process the poisons and get them out of the system but other organs can also be at risk. It depends on what the poisons are and what they do. A chicken with a damaged organ will not necessarily fall over dead immediately. It will not thrive as well as an undamaged chicken. It may not lay as many eggs. It may be more susceptible to diseases or stress. A lot of time you don't know if you damaged your chicken or not.
An example of moderation. Many of us feed cabbage and cabbage relatives to their chickens as a treat. I certainly do. Cabbage contains a poison to chickens and humans. You have to eat a whole lot of it to get enough poison for it to affect you, but it is possible. There was a news story not that long ago where a woman caused her thyroid to shut down by eating 5 pounds of one of the cabbage relatives every day for weeks. I think it was Bak Choi but not 100% sure. Im not eating 5 pounds a day myself and I am not feeding the equivalent of 5 pounds a day to each of my chickens. I think this is good to known but I do not consider cabbage a threat to my chickens.
I selectively throw kitchen scraps and garden debris on my compost pile. I try to keep noxious weed and grass seeds and diseased or infected plants out of it. I don't worry too much about the "forbidden" stuff but I am aware of it. For example, I throw old bean vines with some dried uncooked beans still on them in the compost pile. When I process dried beans I do not throw the rejects on the compost pile because I consider that a concentration that might (not absolutely, every time, without a doubt, bet the farm on it for sure, but might) cause a problem. Use a little knowledge and judgment and don't over worry about it. They are pretty tough critters.
https://www.backyardchickens.com/web/viewblog.php?id=2593-Treats_Chart