I'm referring to the keyword tags, not the html tags. For example: I added the SEO tag to this question.
I'm sure tags are very important for SEO on the question page, but are they also important on the home page?
If StackExchange was to remove the tags from the home page would this have an impact on SEO?
Edits made -- Okay, after raising a few eyebrows, squinting, and re-reading your question several times, I think I know where the confusion is. My original answer still applies to your original question. I am talking about the keyword tags displayed on StackExchange sites along the sides and content of pages, not HTML tags that affect SEO and structure a page in the "view source". These two meanings of "tag" combine at the end of my answer because search engines use HTML tags to know what the keyword tags are... I have attempted to be more clear here...
Original answer:
Yes (to your second question). Keyword tags link related content together and, if implemented into the design correctly, can give search engine context-specific clues about the content on each page and their relationship and value to one another.
Homepage content is, I believe, weighted more in general (at least the area in the first 800x600 px is). The homepage Stack Exchange sites do so well because of their proper use of <h1>...<h6> HTML tags, placement and flow of content, and contextual linking of related, relevant results. And because the content changes frequently, search engines like Google can know it's a reliably up-to-date website.
Take away the keyword tags and the site would still be indexed, but would lose great relational value to both users and search engines. In the end, (at least on Stack Exchange), it looks like the keyword tags are just <a> HTML elements, but being so close to the main article/content as they are provides a substantial value, compared to links off the side of the page or at the bottom, disconnected to the relevant content.
AFAIK, the rel="tag" attribute, or the rel-tag microformat, currently isn't actively used by any search engines as yet. So the only SEO effects they have would be on the internal backlink structure.
That said, using microformats and other semantic web technologies is itself a type of forward-thinking SEO that puts in place—ahead of time—the tools that search engines could use to better organize and rank webpages in the future.
The tag rel attribute makes a lot of sense, since it's visible meta data, so it's less prone to abuse by spammers and blackhat SEOs (of course, they can still be hidden by CSS, but it's still better than meta tags), and it also tells search engines something very important about the content it's associated with, as well as the actual tag page it links to.
The more semantic data you provide to search engines, the better they can understand your content and classify it correctly and help connect it to users who are looking for it. So even if Google doesn't use a particular microformat right now, it doesn't hurt to be proactive in adopting it and letting Google catch up to you instead of always scrambling to change your site each time Google introduces a new ranking factor.
