As with all rodents, Chinchilla's teeth grow throughout their lives, including their front teeth and their molars. From a "survival in the wild" perspective, this constant growth is required to ensure that chinchilla's teeth don't get worn down to nothing from all the chewing their food requires. As pets, we need to make sure our chinchillas have many items to chew on plus their food.
Maintaining Tooth Health with Food
Chinchillas innately know that they need to chew on things continually to keep the growth of their teeth in check. The first step is to ensure that they have teeth-appropriate foods. One example is timothy hay which is quite coarse and reportedly popular with chinchillas and their owners. As a backup, we've heard that high quality meadow hay is also good. Check the hay to make sure it doesn't smell musty or dusty.
Chinchillas should be given as much hay as they want. Unlike with treats, chinchillas won't overeat hay so it's safe to keep a constant supply in their cage. It also makes sense to keep the hay in a rack of some sort attached to the side of the cage to keep it from getting soiled. Hay that has been compressed into cubes is also a good way to keep it from being inadvertently used as bedding.
Other Ways to Maintain Tooth Health
In addition to the right food, you'll want to make sure that chinchillas have enough other things to chew on. Of particular importance is supplying them with safe wood. Cardboard boxes are also good and your chinchilla will enjoy being able to hide in them. For those with a little extra money to lavish on their chinchillas, there are various chew toys available, many made out of wood.