Tires leak oils as they break down. They also emit fumes. the fumes could possibly be hazzardous to your chickens lungs.
The hazardous chemicals chemicals from old tyres are Dioxins, Furans, Nitrogen oxides, Sulfur oxides, Lead and Chromium compounds. These are potential hormone disruptors, and can cause fetal deformities...
I personally would not want these around my laying hens, especially if I planned on hatching eggs. I just would not want to risk it.
Instead, I'd clean the tires with soap and water and turn them into raise flower bed planters... wash them, lay them where you want, filll and cover with dirt/soil, and then fill with flowers... I have seen people use them for potatoes, but I wouldn't do that either... would be too afraid of contamination of my food.