それを行う1つの方法
SELECT a.id_article, a.name
FROM web_article a
WHERE NOT EXISTS
(
SELECT *
FROM web_tag_to_article ta JOIN web_tag t
ON ta.id_tag = t.id_tag
WHERE id_article = a.id_article
AND t.name = 'News'
)
また
SELECT id_article, name
FROM web_article
WHERE id_article NOT IN
(
SELECT id_article
FROM web_tag_to_article ta JOIN web_tag t
ON ta.id_tag = t.id_tag
GROUP BY id_article
HAVING MAX(t.name = 'News') = 1
)
また
SELECT a.id_article, a.name
FROM web_article a LEFT JOIN web_tag_to_article ta
ON a.id_article = ta.id_article LEFT JOIN web_tag t
ON ta.id_tag = t.id_tag
GROUP BY a.id_article, a.name
HAVING MAX(COALESCE(t.name,'') = 'News') = 0
これがSQLFiddleのデモです