Having multiple H1 header tags on your website could be causing an issue with your individual pages. Stick to keeping your website and its pages WC3 compliant with proper use of the H tags. It makes the page easier to read and if needed you can style the H tags using a CSS style sheet.
The H Tag flow is making use of the H1, H2, H3 and H4 tags in the correct order and format. Whilst not an important factor in ranking terms it could well be used as one of the ranking factors that Google uses to place a we sbiet in its position in searches.
There should only be ONE H1 tag on each page, anymore and it may be an issue of bad coding and W3C Compliancy. Here is the correct layout and order on a web page to use for header titles:<h1></h1> - One per page
If you keep the header title tags in the correct layout order the H Flow through your pages will be better for it. Make sure you have your main keyword in the H1 tag and that you don't use it in every H tag on your page, that's overkill and may be classed as keyword text spamming.
Will it increase the ranking positions of your website? On its own probably not but its just one of a quite a few factors and all together they will make a difference and all factors need to come together to make the website compliant to Google and website visitors usability.