I have never heard of a leaser purchasing insurance for a horse they do not own. Insurance for yourself? Yes.
I have both my mares insured through my own insurance. I also have my saddles insured. I pay once per year (can't remember how much), and it insures them against barn fire/trailer accident/death due to surgery (I think). If any of those were to happen, I would be payed their worth.
I think that would honestly be something that is up to they owner to pay, unless the horse/tack/what have you was already insured and the owner has included that cost in your lease fee. Just my 2 cents.