You can use Delete Features to delete all or a selection of locations, just as you would for a normal layer:
Delete Features—Help | ArcGIS Desktop
The trick is how to access the sublayer (like Stops or Point Barriers). In Model Builder, you can use the Select Data utility to grab the sublayer from the network analysis layer. In python, the syntax depends on whether you're using ArcMap or Pro, but it involves doing listLayers on the network analysis layer object. The last code sample on this page shows it being done (for ArcMap):
Make Service Area Layer—Help | ArcGIS Desktop
If you're going to add a completely different set of stops or barriers to your sublayers after deleting the existing locations, you actually don't need to do a delete step at all. When you run Add Locations, just chose the option to clear existing data rather than append.
