The sign above is one of the best that I've seen at explaining the danger of poison oak. The photos are to show you what it looks like at two different times of the year (spring photos are above and below and fall is in the middle) and to give you some useful descriptive language.
As the sign says, sometimes the plant looks dead or woody, but it's still toxic to humans. You can get a rash (**Warning: graphic photos) from the oil urushiol in the leaves or stems of this plant. Although your dog may not be as susceptible to rash, if it runs through poison oak, you, as the pet owner, may get exposed to the toxic oil by touching your dog's fur.