I would appreciate an answer specifically in the form of an exothermic chemical reaction. Namely, the one primarily responsible for generating heat in warm blooded animals that does not take place in cold blooded animals.
Brown adipose tissue or brown fat is one of the primary ways of generating body heat, and it is only found in warm-blooded animals. It is brown due to the high numbers of mitochondria, and heat is generated by uncoupling the electron transport chain from ATP synthesis by oxidative phosphorylation.
In the typical (ATP-producing) mitochondria, an energy gradient is formed by storing protons within the inter-membrane space, and these then flow through the ATP synthase to power the generation of ATP from ADP and phosphate. In brown fat mitochondria, Uncoupling Protein 1 (UCP1 or thermogenin) forms a pore in the inner membrane, allowing the proton/electron ratio to reach equilibrium and robbing ATP synthase of its motive power. This is what generates the heat.
While all animals and even plants generate heat from chemical reactions or mechanical motion of their tissues (like muscle tissues), warm blooded animals have brown fat which has the unusual ability to generate heat directly from metabolic energy.
The particular chemical process you are looking for is performed by uncoupling proteins (UCPs). They reside in the mitochondria and rather than use the proton gradient in the mitochondria to generate ATP, UCPs pass protons through the membrane and generate heat.
Brown fat is brown because of a relatively large iron content associated with many extra mitochondria (brown adipose tissue appears to be derived from muscle cells and not white adipose tissue). Its a heat generating organ in humans, particularly infants.