This principle is so fundamental. so basic that either you are dealing with a novice investor type or you do not fully understand the situation.
The answer to your question is contained in the lease/purchase agreement.
If the agreement does not state buyer is required to purchase home insurance AND buyer is required to pay property taxes....that is your answer. An owner who delegates this to an buyer/tenant is not receiving good advise.
If so, there should be a contract that everyone Is going by. This should be spelled out in it. You should probably find a good real estate attorney to advise you on the matter.
Best of luck,
Regardless of your agreement - if there is no insurance in place, and something happens, it's a bad thing. If the property tax isn't paid and the county forecloses, it's a bad thing.
If you think the other guy should pay, and they haven't, you better pay, because if something bad happens, it will happen to you.
All the best,.