Are there any plants or trees that can live directly on soil that has fresh chicken manure?
One suggestion might be Couch grass. I make this suggestion for the following reasons: applying "hot" materials to anything with a light fibrous root system will have an enormous effect on the root hairs and if severe enough will kill the plant. A similar "hot" material is wood ash or human liquid waste, both of which in undiluted form will kill many plants. The problem is the high acid or high alkaline nature or high salt of the material which will stop root hairs functioning.
One solution might be a plant with a thick rhizomatous or tuber like root structure which can suffer the temporary effect of root hair death and have sufficient resources to put out more root hairs as the "hot" material is modified by soil bacteria into a form that the plant can use as food.
One such candidate is couch grass if it is suitable for your soil. It can form a thick mat several inches deep and seems to persist with great vigour despite being roughly treated. While applying a thick layer of fresh manure might cause the top growth to slow due to lack of light, once the manure rots sufficiently to allow the light in leaves will grow again. It takes a couple of years to discourage couch grass with landscape fabric, and that is lots of time for manure to rot.
I think many plants can live with much manure.
Chicken manure is a very good manure. I think it is better then horses' one), and it is used for vegetable garden. If you have foxes and other chicken killer animals, I recommend you to dig it, not to attract them (and they will dig in your vegetable garden to look for rest of chicken).
Tomatoes like them particularly. (But now I don't remember anymore what other plants prefer it from other manures).
It's important to consider why a plant might not grow in it, and then think of plants that could. Looking at another answer, the word "hot" is particularly unhelpful. Assuming the wood ash is not still hot from the fire, its damaging effects are due to the extremely high pH being slightly caustic and destroying root hairs. In terms of manures, there are two reasons why they might damage a plant. Firstly, when actively composting microbial action can generate high temperatures which literally cook plant materials. This can be avoided by waiting until temperatures have cooled or by spreading the material thinly to help it cool. The second way in which they can damage plants is through nitrogen scorching which has nothing to with temperature but is to do with high concentrations of solutes outside of roots limiting uptake of water.
Fresh chicken manure is particularly rich and so high temperatures can be generated during composting (>60 degrees C). No plant is able to resist prolonged exposure to these temperatures, and hot composting is used to ensure that the resulting material is free from viable weed seeds and roots. If the heap is hot, it might be possible to mound over it with topsoil (depth > 6"/150mm) and so long as soil temperatures are kept in a reasonable range (<35 degrees C) this can make an ideal spot for growing crops such as cucurbits, which benefit from the warmth and supply of nitrogen.
Nitrogen scorch is also likely to be a problem - again mounding over with topsoil would help as will ensuring plants are adequately watered.
If it is not important to use it, you could see which plants colonise it and at what stage the manure is at. This will tell you a lot about the plants growing locally to you. In the UK, Elymus repens may well be one of them, but I've seen Rumex and Urtica species get in on the act pretty quickly as well.
Another option to consider (which is something I do) is mix chicken manure with steer manure (1:3) in the soil where you want to grow the vegetables. Using fresh manure can be a health risk if you are growing vegetables. I use processed manure from big box stores so don't know how you can process your chicken manure yourself. If you mix up the manures as I suggested, you can pretty much grow any vegetable that is conducive to your environment and that you like to consume.