My dog eats hop leaves and roots (if I let him which I don't! bad dog!) to no ill effect. He even got into the mash/wort detritus one brew day and while it gave him terrible gas, he lived though I wish I had died after spending 15 minutes in a room with him.
The story that is often linked to this internet rumor has no corroboration. If you google it you can see it's been around for about 15 years and the number of dogs effected varies with each retelling as well as which doctor/research group that is actually claimed to have published the report. I even found it referenced in a book on herbs but again no evidence was actually sited, just a reference to a University (no link to published paper etc...).
But if you want to err on the side of your dog, by all means don't let them eat hops. It's not like they need them to be healthy.
Here's the book reference. I found it on Google books.