Quando você usa várias junções na consulta principal, acaba com um produto cruzado de todas as tabelas, de modo que as somas são multiplicadas pelo número de linhas correspondentes em outra tabela. Você precisa mover as somas para 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
