Change his filter and gasket and he's good for another 100K miles.

Sorry, couldn't help it.

I imagine that since it's a petroleum distillate that it is poisonous to some degree. Also, dog skin is pretty sensitive. The dog has hair to protect it so it has a thin epidermis, which means live tissue is near the surface, easily irritated.

I'll bet a very high lanolin content cream would help. Lanolin is a natural skin oil, so it is very unlikely to further irritate the dog's skin. I'd stay clear of medicated creams, the skin is already likely to be irritated. Also, I'd go easy using detergents (shampoo, etc). Use a mild soap followed by a water rinse and then slather on the lanolin. The fluid is already able to dissolve oils to some degree, drying the skin a lot. Using excessive amounts of detergents would dry the skin even more. Instead, I'd use the high lanolin cream to 'clean' excess hydaulic fluid from his hair and body.

The dog is probably going to spend a lot of time licking hydraulic fluid off it's hair, even after you've cleaned. Also lick off excess lanolin. All of that may induce vomiting. Look out.