Maybe not for search engines (as much), but it can confuse users if you use more than one per page. Especially those users who have to use assistive devices such as screen readers.
<h1></h1> is a pair of tags that contain text which is used to mark up a block as a "level 1 heading" or "top level heading".
Yes, it is the purpose of it like a header on pages in a book or on a news paper. I use <hx tags more then ones on a page if the content requires it.
Using H1 and then the other H tags in descending order is a prerequisite of semantic markup. It also helps the search engine bots determine what your page content is about, as has been mentioned. All the people who are saying only use it once, what are you basing that on? Has anyone tested it?
A good rule of thumb is to use each h tag only once per page. Use the h tags for headers ie sections that provide a brief summary of that page or that section of text. Inbetween your h tags use paragraph / <p> tags.
The numbered once should only be use once: For example: Main header: <h1> Sub-heading: <h2> Introductory text <h3> It helps a bit with formatting and editing. And here some more reliable guidelines would be required.