Here is a post from my team mate Brian Schulman. It's the same advice I give to my tenants.
In Lancaster, PA this week, a family was burned out of their home. They lost everything. As a landlord, one of the things my leases have specified is that the tenants must have insurance on their personal possessions, as the landlord's insurance only covers the buildings and not their personal property.
I make a point of advising renters to get replacement value insurance for their things, rather than actual cash value (ACV). The example I use is: let's say you lose your television in a fire. Let's say the TV is 6 years old. You will have to get a replacement - and you'll probably be buying a new TV. If you're only insured for ACV, you'll only receive what that TV would be worth in a garage sale or re-uzit shop. Think about what your used clothes would be worth now - and what you'd have to pay to replace them! Plus, the difference in premium cost between the the two coverages is really minimal.
There's a parallel in insuring older homes in low-priced areas of cities. Often it would take quite a bit more than a home is worth to replace it in the case of a total loss. Many insurance companies have a minimum coverage amount that they will require regardless of the actual lower resale value of the home. Although it may be possible to insure for only the amount owed on the mortgage, that may not be wise. In case of a loss, rather than ending up with just the mortgage paid off and an empty lot to sell, the owner may be considerably better off rebuilding - but that may only be possible if adequate replacement coverage is in place!
Another consideration is that at the same time you are protecting the interests of your tenants, you are also reducing your own liability as a landlord in case of a loss, by showing that in the lease you required your tenants to carry insurance on their private property..
A legitimate place to lower insurance premiums may be by adjusting the deductible. Many people think nothing of having a $1000 deductible on their $10k to $20k car. Why then should they buy a $200,000 home and insure the home with a $500 or lower deductible? Most people would not make an insurance claim for less than $1000 anyway - so why unnecessarily increase your premiums?