It would depend on the dog and the chinchilla if it will get stressed out or not just having a dog there. So long as the dog is calm and doesn't sit there focused on (staring, whining, barking) the chin, or try to get the chin it should get use to the dog in time. I know some people put fencing around the cage so the dog can't get right up to the cage. Keep in mind that a dog's mouth has a lot of stuff that can make a chin sick, so no licking the cage, and especially no licking the chin. If the dog is sneezing or sick it shouldn't be in the same room until it's better.
I have several cats and a couple dogs that spend time in the room with the chins, my chins are fine with them just hanging out and walking by. My other dog though, I can't even have in the same room because he has a very high prey drive and is very intensely focused on the chins. So know your dog, is she good with other animals and knows to leave it and give things space when told? Or is she more the type to want to stick her nose in everything, or has a high prey drive or energy level. Also does your dog bark a lot, that can scare the chinchilla too.
It's not so much that chins don't get along with other animals so much as other animals can be dangerous for the chins to be around. They have small thin bones, they may look big but are really only about half the size under all that fur. Other animals can accidentally hurt or kill a chin in play. They can also make them sick with bacteria they can carry. For example pasteurella, which can be deadly to chins (it causes pneumonia), is a normal bacteria in dogs and cats noses and mouths, and can be carried by rabbits without symptoms.