From what I have gathered, Tags don’t directly affect your SEO ranking – keywords put you “on the map”, but really it’s your content and popularity that will increase your ranking.
However, many people rave that tags (properly [not overly] done) have a great affect on flow through your website.
Example: You write a FANTASTIC! article about puppy chow. I read your article and say “MY GOD, this guy is A GENIOUS when it comes to puppy chow!”
After reading your article, I see that there are two tags at the bottom of the page – “dogs”, and “pet food”. Well I immediately click on “dogs” to see what else you have to say on that topic.
Now, that may have been a ridiculous example, but I am sure you got through it easy enough. Tags create traffic, and traffic increases your SEO ranking. So having useful (and not overdone) tag navigation will increase traffic flow on your site, reduce the number of people who come for one article and then leave (we all hate that, don’t we?).
As for direct ranking results, no, having “dog” tagged on a post will not make it appear any higher than it will be with no tags. However, that “dog” tag may bring in 5, 50, 500 more readers from other “dog” articles on your website, which will tell Google “Hey, this guy knows what he’s talking about, he just got 500 views on a single post”.
Hope this helps!