It's hard to say without knowing your schema, but something like this will work:
select article_id from articles
inner join tag t1 on t1.article_id=articles.article_id and t1.tag='included_tag'
inner join tag t2 on t2.article_id=articles.article_id and t2.tag='another_included_tag'
left outer join tag t3 on t3.article_id=articles.article_id and t3.tag='dont_include_tag'
left outer join tag t4 on t4.article_id=articles.article_id and t4.tag='also_dont_include_tag'
where t3.tag_id is null and t4.tag_id is null
inner join with tags that should be included, and do anti-join (outer join + where the required column is null) tags that shouldn't be included