How can I hide a column using a Select statement?
You just need to add a case statement to your select list:

declare @bid varchar(4) = '1007'    

select sr.release_date,
    sr.customer_id,
    sr.customer_name,
    sr.item_id,
    sr.item_desc,
    sr.order_no,
    sr.qty_on_hand,
    sr.release_qty,
    sr.qty_after_released,
    sr.extended_sell,
    sr.enough,
    case when sr.route_code in ('pm 01','pm 03','pm 05') then sr.route_description else '' end as route_description
from sri_schedule_releases_view as sr
where 
    ((case 
        when sr.route_code in ('pm 01','pm 03','pm 05') then (select users.contact_id from users where users.delete_flag = 'n' and users.name = 'John')
    end = @bid) or 
    (case 
        when sr.route_code in ('pm 01','pm 03','pm 05') then (select users.contact_id from users where users.delete_flag = 'n' and users.name = 'Jane')
    end = @bid) or
    (case 
        when sr.route_code in ('pm 01','pm 03','pm 05') then (select users.contact_id from users where users.delete_flag = 'n' and users.name = 'Doe')
    end = @bid) or
    (sr.contact_id = @bid))
order by sr.item_id asc