While there is not currently a way to permanently delete a customer from Heartland Retail, you can mark the customer as "inactive" in order to prevent them from appearing in future searches.
In order to make a customer inactive, first pull up the customer by clicking on Sales in the top navigation menu, then select Customers. Type the desired customer name into the search box and select the customer name. In the top right corner there will be a green switch that says "active."
Select the switch and the status will change from "active" to "disabled."
The customer is now inactive!