Can lime sulfur dip harm my pet? What are the dangers and risks?

Lime Sulfur dips do carry some risk and the MSDS link here shows some of the risks with its use. https://davismfg.com/resources/uploads/Lime%20Sulfur%20Dip.pdf

The suspected deaths that I have recovered in the veterinary files indicate problems arising when the pet accidentally ingests some of the liquid causing corrosive of the GI lining, accidental aspiration of the liquid down the trachea causing pulmonary edema. Also I have commonly seen user error where the applicant didn't dilute the dip properly causing superficial burns.

So yes it can be dangerous if any of these things occur. Which is why I generally don't recommend Lime sulfur dips as the first choice of treatment because owners don't always read the safety precautions on the bottle and abide by them, and sometimes accidents do happen. A safer treatment for mites is either oral Bravecto, Nexgard, or Ivermectin 3% (which in itself carries some risks as well that should be discussed with your veterinarian).