My perspective on insurance is that insurance should be taken out for cases where you insure against a large, unlikely risk that you could not afford to cover, and the risk is not specifically knowable. So insuring your house against fire is appropriate, you would not afford a $200,000 or $500,000 loss. (By "specifically knowable" I mean a house fire may be caused by a random accident that you had no control over).
On the other hand, there is insurance say for accidental damage to a mobile phone. You should be in a position to cover the cost of a replacement phone. From an insurer's perspective, accidental damage to a phone is on average a very likely risk, so you would be paying maybe 25% or more a year of the value of the phone as premiums. Similarly, when I had a car worth less than £1000 I was happy to have third party insurance only (insured against damage to others and for theft of your vehicle, but not insured for accident damage to your vehicle), it was not worth paying more to insure less than a £1000 risk. So, yes, self-insure on smaller risks.
The total value of my possessions is probably around GBP £1000 for electronics and under GBP £1000 for clothes. So for that I happily self-insure as a renter.
As as renter in the UK, the landlord has to insure the building (and optionally his furniture), the tenant does not have to have any insurance and isn't liable for fire damage etc. even if their fault by accident (obviously criminal arson is another matter). So as a UK renter no insurance necessary, but as understand it in the US the rules may be different. If as a renter you could be liable for the cost of the building in the event of fire, then yes I would insure against that.