What organ systems would be in involved in homeostasis or thermoregulation?
First, it must be clarified that homeostasis and thermoregulation are not synonymous. Homeostasis is a term used to describe an organism's ability to maintain a stable and balanced internal environment. In terms of animals, there are many physiological variables that must be maintained within set range. This includes things such as pH, hormone levels, and temperature.
An animal's ability to thermoregulate is a form of homeostasis. If discussing warm-blooded animals or endotherms, one system involved would be the muscular system which will contract and relax rapidly (shiver) for heat production. In addition, the integumentary system is partially comprised of a layer called the hypodermis which contains adipose tissue. This tissue helps to insulate the body. Sweat glands are accessory organs to the skin and they will release sweat. Once that sweat begins to evaporate, excess heat will be carried away with it.
If discussing ectotherms or cold-blooded animals, they must thermoregulate via the enviroment. Because of this, their integumentary system is the main system involved in thermoregulation. For example, if a snake wants to "cool off", it will relocate to shade or water. If it wants to "warm up" it will relocate to a sunny spot. Once their skin takes on the temperature of the environment, it will continue to be transfered internally.