My sister's apartment just burned down due to a faulty space heater in the greenhouse/ florist beneath her, which is the owner's business, and apartments. Sister had no insurance. The landlord will be reimbursed, will they also reimburse the tenants, since they burned them out through negligence?

No. Your sister should/could have had her own renter's insurance to cover her apartment's contents. I don't see the landlord's insurance covering her contents. Renters insurance is relatively cheap each month usually less than $25 and even less if combined with the auto policy with the same company. Even if no auto, the rate is still inexpensive considering what it costs to replace the contents.

She can sue her for the value of the damage under the liablity, if it is proven that something the owner did caused it.

If it was merely a problem with a piiece of equipment it may not be

For the landlord to be liable, it would have to be shown they were negligent. If they knew the heater was faulty and still used it or used it improperly, they could be held liable. But its unlikely your sister can prove that. She has probably learned an important but expensive lesson. No homeowner would think about not getting insurance on their home and no renter should either.