When any is used as a pronoun in questions to refer to countable nouns, it can be used with either singular or plural verbs: is any or are any. When you use a plural verb (like are any), you are asking about “one or more” of a number.

Should I add s after any?

If you use any or some with countable nouns, the nouns should be plural, i.e. you should add an 's'. If the noun following any or some is uncountable, don't add an 's'.

Do any or does any?

It would be correct to say "Do any philosophies believe" because now you have a plural subject, "philosophies," and a plural verb form. It is confusing that plural nouns typically end in 's' whereas singular, third person verbs in the present tense end in 's."