Cats and dogs do not suffer the allergic reaction to skin contact with urushiol that we humans do. What about ingestion?
Imagine the following hypothetical1 near-worst-case2 scenario: A human has walked through a patch of poison ivy, creating direct contact between the leaves of the plant and his shoes. Further, this human steps in a small pile of rabbit pellets. A few minutes later, he enters his home and kicks off the shoes. Thereupon a cat, lured by parfume du bunné, licks the shoes.
What health risks are there for said cat?
1: If you read the edit history of this question, you will note that it is motivated by a real situation involving a very low but non-negligible probability of ingestion. However, I realized overnight my actual question is about the hypothetical, not the real.
2: Of course, the full-on worst-case is a cat actually eating a poison ivy leaf. If an answer wishes to address this, go for it, but I think second-hand ingestion as described is more likely.