SQL Server 2019

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WITH yourTable AS (
    SELECT 1 AS PersonID, '2012-04-26' AS Visit_date, 1 AS Medication_intake, NULL AS Previous_date, NULL AS Date_diff UNION ALL
    SELECT 1, '2012-11-16', 1, '2012-04-26', 204 UNION ALL
    SELECT 1, '2013-04-11', 0, NULL, NULL        UNION ALL
    SELECT 1, '2013-07-19', 1, NULL, NULL        UNION ALL
    SELECT 1, '2013-12-05', 1, '2013-07-19', 139 UNION ALL
    SELECT 1, '2014-03-18', 1, '2013-12-05', 103 UNION ALL
    SELECT 1, '2014-06-24', 0, NULL, NULL        UNION ALL
    SELECT 2, '2014-12-01', 1, NULL, NULL        UNION ALL
    SELECT 2, '2015-03-09', 1, '2014-12-01', 98  UNION ALL
    SELECT 2, '2015-09-28', 0, NULL, NULL
),
cte AS (
    SELECT *, ROW_NUMBER() OVER (PARTITION BY PersonID ORDER BY Visit_date) rn1,
              ROW_NUMBER() OVER (PARTITION BY PersonId, Medication_intake ORDER BY Visit_date) rn2
    FROM yourTable
)

SELECT PersonID, Visit_date, Medication_intake, Previous_date, Date_diff,
       CASE WHEN Date_diff IS NOT NULL AND Medication_intake = 1
            THEN SUM(Date_diff) OVER (PARTITION BY PersonID, rn1 - rn2
                                      ORDER BY Visit_date) END AS Running_sum
FROM cte
ORDER BY PersonID, Visit_date;

PersonID	Visit_date	Medication_intake	Previous_date	Date_diff	Running_sum
1	2012-04-26	1	null	null	null
1	2012-11-16	1	2012-04-26	204	204
1	2013-04-11	0	null	null	null
1	2013-07-19	1	null	null	null
1	2013-12-05	1	2013-07-19	139	139
1	2014-03-18	1	2013-12-05	103	242
1	2014-06-24	0	null	null	null
2	2014-12-01	1	null	null	null
2	2015-03-09	1	2014-12-01	98	98
2	2015-09-28	0	null	null	null

Warning: Null value is eliminated by an aggregate or other SET operation.