Cuando usa combinaciones múltiples en la consulta principal, termina con un producto cruzado de todas las tablas, por lo que las sumas se multiplican por la cantidad de filas que coinciden en otra tabla. Debe mover las sumas a las subconsultas.
SELECT last_name, first_name, DATE_FORMAT(LEAST(mil_date, tm_date), '%m/%d/%y' ) AS dates,
total, minutes
FROM bhds_teachers AS i
LEFT JOIN (
SELECT ds_id, YEARWEEK(mil_date) AS week, MIN(mil_date) AS mil_date, SUM(drive_time) AS minutes
FROM bhds_mileage
WHERE mil_date BETWEEN '2016-04-11' AND '2016-04-30'
AND bhds_mileage.ds_id = 5
GROUP BY ds_id, week) AS m
ON m.ds_id = i.ds_id
LEFT JOIN (
SELECT ds_id, YEARWEEK(tm_date) AS week, MIN(tm_date) AS tm_date, SUM(tm_hours) AS total
WHERE tm_date BETWEEN '2016-04-11' AND '2016-04-30' AND bhds_timecard.ds_id = 5
GROUP BY ds_id, week) AS t
ON t.ds_id = i.ds_id AND t.week = m.week