Who is responsible for damage caused to a tenants personal property when the ceiling collapsed during a storm because the building was not weatherproofed correctly and the tenant didn't have insurance?

The answer is basically no. The landlord is never responsible for damages to personal property belonging to tenants in any dwelling or structure. That is why you are strongly encouraged to get renters insurance. In some cases you may have some recourse if the landlord knew of the problem. But if you get renters insurance be insurance company can determine that for you and they can seek damages from the landlord.