Chrysanthemums contain pyrethrins and sesquiterpene lactones that irritate the GI tract and skin.
The same pyrethrins are used in flea sprays, where they are toxic to cats — they are mildly toxic to dogs too.
Chrysanthemums contain pyrethrins (natural insecticides) and sesquiterpene lactones. In dogs these cause vomiting, drooling, ataxia and skin irritation. Concentrated pyrethrin/pyrethroid products are far more dangerous than the plant itself.
Sources
Eating a few leaves usually causes GI upset; large amounts can cause neurological signs.