Should a teaching statement always be included in faculty job applications, even if not required? Should one always write a teaching statement when applying for faculty positions, even when not explicitly asked for in the job post?

If it depends on the field and/or type of university, please specify. If it matters, I'm applying to Physics departments in universities which primarily emphasize research, which already require (at least) a CV, research statement, cover letter, and letters of recommendation.
I don't think you need to include a teaching statement if one is not explicitly requested.  However, you should be very clear in your application materials about what courses you want to teach and/or are willing to teach.  That is usually the information we are looking for when we ask for that document.

Including a teaching statement when one is not requested is probably not going to be a negative (most applications we get have just been sent out in bulk, and many places require one) unless someone on the committee is exceptionally nit-picky.  But if that is the case, do you really want that person as a colleague?