If your chinchilla's fur begins falling off in patches, has scabs on different areas of its body, & may be redish in color it may have mites or fungus. If the animal is always chewing or scratching at an area of its body check it out. One advantage to their beautiful fur is they cannot get fleas or lice; their fur is too thick!
Fungus & mites can most times be cured with adding a touch of athletes foot powder to their dust bath & using that a few times a day.
Many chinchillas are exposed to the mites or fungus from being exposed to an infected animal or transferred from your hands after touching something infectious. That is why all the pets in your house must be treated with the proper medication. You must also sterilize the chinchilla's cage/bedding, as well as any other caged animals. Make sure you wash your hands thoroughly between treatments & when your done to prevent from spreading.
If irritation continues or your chinchilla seems to be in pain, dont question it & get it to a Vet!