Google's John Mueller said that your site can rank fine with or without H1 tags. You don't have to use any H1 tags, or you can use many, either way, it is not a critical issue specifically for Google search.

He did say it helps Google understand the structure of the page, just like it helps users understand the structure of the page.

He said this in a video this Friday at the 23:47 mark when he was asked if "it is mandatory to just have one h1 tag on a web page or can it be used multiple times?"

He responded:

So we we get this question multiple times as well. You can use h1 tags as often as you want on a page. There's no limit. Neither upper nor lower bound. h1 elements are a great way to give more structure to a page so that users and search engines can understand which parts of a page are kind of under different headings. So I would use them in in the proper way on a page. And especially with html5 having multiple h1 elements on a page is completely normal and kind of expected. So it's not something that you need to worry about. And some SEO tools flag this as an issue and say like oh you don't have any h1 tag or you have two h1 tags. From our point of view that's not a critical issue. From a usability point of view, maybe it makes sense to improve that. So it's not that I would completely ignore those suggestions but I wouldn't see it as a critical issue. Your site can do perfectly fine with no h1 tags or with five h1 tags.

Here is the video embed:

Forum discussion at YouTube.