t, PostgreSQL - rextester

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CREATE TABLE Table1
    ("id" int, "date_start" timestamp, "date_end" timestamp)
;
    
INSERT INTO Table1
    ("id", "date_start", "date_end")
VALUES
    (1, '2017-02-03 08:00:00', '2017-02-03 17:00:00'),
    (2, '2017-02-04 15:00:00', '2017-02-05 10:00:00'),
    (3, '2017-02-06 14:00:00', '2017-02-07 23:00:00')
;

WITH days as  (
    SELECT a.n
    from generate_series(1, 100) as a(n)
), ranges as (
    SELECT *, (d.n::text || ' DAY')::interval as i,
           t1.date_start::date + ((d.n - 1)::text || ' DAY')::interval as new_start,
           t1.date_start::date + (d.n::text || ' DAY')::interval as new_end,

CASE WHEN t1.date_start::date = t1.date_end::date AND d.n = 1
                THEN t1.date_start    
                WHEN t1.date_start::date < t1.date_end::date 
                THEN t1.date_start
                ELSE NULL
           END as date_start1,
           CASE WHEN t1.date_start::date = t1.date_end::date AND d.n = 1
                THEN t1.date_end
           END  date_end1
    FROM Table1 t1
    CROSS JOIN days d
)
SELECT *, CASE WHEN date_start < new_end AND date_end > new_start
               THEN 'overlap'
          END as overlap,
          CASE WHEN date_end1 IS NOT NULL 
               THEN date_start1
               WHEN date_start < new_end AND date_end > new_start
               THEN CASE WHEN date_start > new_start
                         THEN date_start
                         ELSE new_start
                    END
          END as final_start,

CASE WHEN date_end1 IS NOT NULL 
               THEN date_end1
               WHEN date_start < new_end AND date_end > new_start
               THEN CASE WHEN date_end < new_end
                         THEN date_end
                         ELSE new_end - '1 second'::interval
                    END
          END as final_end                      
FROM ranges
WHERE date_start < new_end AND date_end > new_start
ORDER BY "id", new_start


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