The <H> tags on you page should be used to detail the important content on the page. Think in terms of a book.
<H1> is the title (Only one)
<H2> is a chapter heading (multiple)
<H3> is a subsection (multiple)
etc

Use CSS or inline styles to alter the look of the content described by the <H> tags.

Multiple <H1> elements will be a problem.

Thanks for the feedback

Can a search engine consider this overuse, and penalize me?

Not really. But, if you take that and then look at other areas, if there is a pattern, then you may be in for a rough ride. I really don't think the algo is going to do anything (yet), but a manual review might raise some flags if there are multiple signals present.

For example, you might have multiple <h1> elements. While the guidelines specify one <h1> per page, many don't know that guideline so it is something that could be overlooked. But, if I traverse through the code and find that alt attributes have been abused and are assigned to every single image including spacers, bullets, etc., then that's probably going to get you in a little deeper. Then, if I find that anchor text is extremely repetitive, you're probably going to get one of those colorful little buttons in my Quality Reviewer Control Panel. They are "cautionary" colors so they do not work in your favor.

I've seen entire pages wrapped in an <h1> element and find their way into the top results. Someone's idea of adding more relevance to the entire page I guess. At that time, I really don't think semantics were part of the mix. It was strip out all html and what you're left with is what is crunched. I think it goes a little bit further these days. Technology has improved...

Don't forget to keep each page's H1 tag unique. I used one template and it used thesitename .com as the H1 on every page, and the page heading was H2. Needless to say I changed that.