The subject in both sentences is "philosophy," which is singular.
The verb in both sentences is in the present tense. Singular, third person verbs in the present tense (except for modals) terminate in "s" or "es." Subject and verb must agree in number. Therefore your first example "do any philosophy believe" is not correct whereas your second example "does any philosophy believe" is correct.
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."
You are correct that "believes" would be incorrect with either "do" or "does." When using an auxiliary verb plus an infinitive, it is the auxiliary that is inflected (changed) to indicate person or tense.
Finally, what is perhaps a personal point of style. I do not like "believe" in this context. People and perhaps some kinds of animal may hold beliefs, but a philosophy is not a person. A philosophy can teach or propound propositions, but the philosophy itself cannot believe propositions. I admit that people do say things like "Christianity believes," to mean "Christians believe" or "Christianity teaches." So your use of "believe" is not wrong. It is just not the best word available.