Insurance for owning and renting a co-op?

You can conisder not purchasing insurance as a landlord in a coop. First, check the building's master insurance policy to determine that they have adequate coverage for the building in terms of fire, liability, etc. In coops, the building's insurance policy will cover your apartment in the event of a loss but will not cover personal property which is the coverage you would need if you were to occupy, or in this case, the coverage your tenant needs and that it is their obligation, not the landlords. You might consider liability insurance, and/or also consider structuring the purchase under a LLC as investment property if the Board will allow to help limit your liability in the event of accidents (again, the building policy should cover liability).

Or, a simpler approach would be the following:

(1) check your own insurance policy to find out what you'd be covered for
(2) check your building's bi-laws to find out what the insurance requirements are for renters (when I rented a coop I was required to have a certain level of insurance for liability, homeowners, etc).

(3) Regardless of (1) & (2), you should strongly suggest your tenant purchase insurance, perhaps require it per the lease. Either way they'll be liabile for any damage, per the terms of the sublease agreement, but having insurance really makes sense. And renter's insurance is fairly inexpensive. And it may proclude you from having to purchase insurance while you're renting it out.