"no one do" is incorrect in standard US English. "No one" is singular, and so the singular form "does" is required. "Do" is the plural form, and "does" is the singular form of the verb "to do".

Does and do uses?

“Does” is used for singular subjects like “he,” “she,” “it,” “this,” “that,” or “John.” “Do” is used to form imperative sentences, or commands. Example: Do your homework. “Does” is never used to form imperative sentences.

Do or does with 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."