別テーブルに存在しないレコードだけを引っ張ってくる。

SELECT DISTINCT no, name_sei, name_mei
FROM user
WHERE	class = ?
AND	no NOT IN
(
	SELECT no__user
	FROM examinee
	WHERE 2 = flgPassing
)
ORDER BY name_sei, name_mei
LIMIT 0, 5

別件（忘れないように）

SELECT DISTINCT no, name_sei, name_mei FROM user WHERE class = ?
AND no NOT IN
(
    SELECT
    examinee.no__user
    FROM examinee INNER JOIN exam_detail ON examinee.no__exam_detail = exam_detail.no
                  INNER JOIN exam        ON exam_detail.no__exam     = exam.no
    WHERE examinee.flgPassing = 2
    AND   exam.class          = ?
)
ORDER BY name_sei, name_mei