Google are going to want to see semantic markup so they can find out blindly what the page is about. The benefit of header tags to push you up the rankings is negligible (too easy to spam). However, if an element of the page is clearly the title, then using a header is good, semantic code.
There is some debate over whether the site title or the page title should get the h1 - with strong reasoning on both sides. To confuse matters, in HTML5, it could now validate if both had a h1.
I remember SEOs used to make me wrap all sorts of non-header content in header tags, which must have made using a screen reader fun. Don't do this - header tags are for marking up headers, and sub-headers. If your page reads correctly without headers, you don't need them. If you are using bold text or divs for headers, you should really be using header tags.
An opinion piece on SEO Moz said this...
This mistake is a little bit more subtle. For years, SEOmoz recommended including
keywords in the H1 of pages. After we started doing formal machine learning correlation
tests we found out that this tactic didn't actually help very much at all (including
the keywords in normal text in bigger fonts worked essentially the same). This was a
shame because it meant we wasted time and energy convincing our clients to update their
SEOs will often struggle on with things that used to work, forgetting that Google are constantly getting wise to their tricks.