Would using an includes for a header have any impact on SEO one way or another? Or would I be better off hardcoding it into each respective page?
It won’t make any difference to SEO. Search spiders only see the output page - there is no way to tell whether a page has used hard-coding, server-side includes, PHP or any other construction method from the coding. And even if there was, they wouldn’t care - it wouldn’t benefit users one way or the other if they favoured sites that were hard-coded over sites that use includes, or vice versa, so they don’t.