Although burdock root is a popular component in herbal remedies for horses being an excellent blood purifier and an excellent healing herb for skin conditions you don't really want burdock in your paddocks.
Whilst not technically poisonous to horses if the burrs are ingested they can cause damage to the upper gastrointestinal tract, irritation of the mouth, tongue and teeth and oral ulcers. Symptoms include excessive salivation, drooling or frothy saliva.
In addition to the annoyance of burdock burs tangled in manes and tails the burs can actually cause cause ocular lesions and "seed" the eyes if caught in the forelock.
Burdock is a biennial weed that can be treated by spot treatment of a weedkiller, the best time to do this is June or July.