This is similar to How to write code to delete individual relationships which covers how to remove a constituent's individual relationship.  Please see the following VB6 code sample which will loop through all relationships for a given constituent and remove all "Organization" relationships from the constituent record and from the organization.  Here we use the COrganization2 object to load the relationship by import ID from the constituent's relationship:
Public Sub RemoveOrgRelationships()

    Dim oRels As CRelationships2
    Set oRels = New CRelationships2
    'Load constituent record using system record id
    oRels.Init REApplication.SessionContext, 753 
    'Find the relationship to be removed and remove it
    Dim oRel As CRelationship2
    For Each oRel In oRels
        If oRel.Fields(RELATIONSHIP_FLD_Type) = "Organization" Then

            Dim oOrg As COrganization2
            Set oOrg = New COrganization2
            With oOrg
                .Init REApplication.SessionContext
                .LoadByField uf_Organization2_IMPORT_ID, oRel.Fields(RELATIONSHIP_FLD_Import_Id) 'Loads the organization relationship

                Debug.Print "Removing " & oRel.Fields(RELATIONSHIP_FLD_Type) & " relationship to: " & .Fields(ORGANIZATION2_fld_NAME)
                'setting to true will also remove reciprocal relationship
                .Delete True

            End With
        End If
    Next oRel
    Set oRels = Nothing
    Set oRel = Nothing
    Set oOrg = Nothing
End Sub

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