If your dog is experiencing respiratory issues, you need to get him to the veterinarian immediately. She will put him on oxygen support either by flow by, placed in an oxygen cage, or in extremely severe cases, he may need to be intubated and kept on oxygen via intubation until he stabilizes.
The symptoms of doTERRA allergies your dog develops will determine his course of treatment. If your dog has a skin reaction, your veterinarian may bathe him with a mild soap or dish detergent to remove as much of the oil from his skin as possible. While some of the oil will already have been absorbed, it will remove what is left on the fur and skin.
If your dog is suffering a runny nose, eyes, or something similar, your veterinarian may again recommend a bath. This will remove as much of the remaining aromatic fragrance as possible. If your home smells strongly of the oil, you may need to air it out as much as possible to remove any lingering scent. Dogs have a sense of smell much stronger than humans so even if you do not smell it, your dog still may.
If your dog is very itchy or has developed hives, the veterinarian may administer an antihistamine to help with these symptoms. You may also have to put a cone on your dog to keep him from scratching or licking certain areas. While you may have sympathy for him and you do not want to do this, it will be very beneficial for him and he will recover much quicker with the cone in place.