In most short sales, the lender will allocate some of the proceeds to pay the property taxes. That includes the delinquent taxes. While the homeowner is technically responsible for the payment of property taxes, they have to make a business decision on whether to pay the taxes. Every dollar a homeowner spends on a short sale property is a dollar that they will not get back.
If the owner pays the taxes in a short sale, then they will not receive any pro-rated credit refund at the settlement. The lender keeps the credit. If the owner does not pay the taxes, they lender will typically pay the back taxes. In other words, if the owner pays the taxes, all of the money paid ultimately benefits the lender. If the owner does not pay the taxes, the lender will end up covering that amount.
If the property is scheduled for a tax sale, in many cases the lender will pay the taxes. A mortgage lender has to be notified of a scheduled tax sale so they are given the opportunity to act to protect their interests.
Most people who need a short sale choose not to pay the property taxes in the hope that the lender covers that expense. The lender usually does.