|I doubt your dog will get a rash. Animals are usually immune to it because of their fur---it protects their flesh from the oils. YOU, on the other hand, are probably not. The oils from the plants can be passed to humans from the animal's fur so your best bet is to not let your dog near it (if possible) or get rid of it yourself. Dress appropriately--thick rubber gloves--the kind you get at Home Depot, etc., rubber boots, long sleeves and pants and be prepared to wash the clothes separately in HOT water--several times, if possible. If necessary have someone undress you so you don't get any oils on your body as you remove your clothing.
An alternative to removing it is to use a tough weed killer such as Brush-Be-Gone. Be careful, though, because this will kill anything else it contacts as well (flowers, shrubs, etc.).