Dogs do not have a reaction to poison ivy or similar plants such as poison oak and poison sumac. However, they can spread poison ivy to any humans who handle them, so they should be washed after exposure.
The painful, itchy reaction that most humans have to poison ivy is due to the effects of an oil called urushiol that the plant contains. Urushiol rubs off on skin, clothing or hair and is quickly absorbed by the skin. Dogs are rarely directly exposed to urushiol due to their fur, but even when their skin does come into contact with it, they do not seem to be allergic. The oil is long-lasting, however, so pets can transmit it to people unless it is washed off.