Any feature that you place on QuickMap is added to the edit layer under the custom tab. To delete a feature you must make the edit layer the active layer and then use the Info tool to select it and then delete it.
If this does not work it could be because you did a network install and the custom.mdb database (the database that your data is saved in) is read only. The location of Custom.mdb can be found by right mouse clicking on the name “Edit Layer” in the layer control tool, and selecting Properties. The database location will be displayed near the top of the form, and is normally C:/Qmap/data/custom.mdb.