Users using certain Roles are seeing the - New - Option from a particular drop-downs.
Users will see the - New - Option in certain drop downs if they have the ability to Create for that type of record.
To remove this function, the permission to create the type of record should be removed from the custom role or the Level should be set to View. This will also prevent users with this custom role from being able to create records for that record type in all other ways.
Note: The Owner of a particular Custom List/Record, will always have permission to Add New Records, thus the - New - Option will always be displayed for the Owner.