How to remove duplicates, which are generated with array_agg postgres function

You can use the distinct keyword inside array_agg:

SELECT ARRAY_TO_STRING(ARRAY_AGG(DISTINCT CONCAT(u.firstname, ' ', u.lastname)), ', ')
FROM log_has_item logitem
  INNER JOIN log log ON log.id = logitem.log_id
  INNER JOIN worker u ON log.worker_id = u.id
WHERE logitem.company_id = 1

SQLFiddle with this example

No need to go all round the houses with ARRAY_TO_STRING(ARRAY_AGG( when a simple STRING_AGG will do as follows (code available on the fiddle here):

--
-- Simplified
--

SELECT
  DISTINCT
  STRING_AGG
  (
    DISTINCT CONCAT(w.firstname, ' ', w.lastname), ', '
  ) AS "The workers"
FROM log_item li
INNER JOIN log l ON li.log_id = l.id 
INNER JOIN worker w ON l.worker_id = w.id
WHERE li.company_id = 1;

Result:

                The workers
Frantisek Smith, Peter Duff