Not wanting the header (thus page) to 'refresh' means not going to another page, and that can only be achieved through ajax afaik.
In that way you'll stay on the same page and your flash animation can loop/do whatever you want, but you'll load in the new content on a certain part of the page. So you'll update only part of the page instead of going to a whole new page.
However, having a site that works only with ajax definitely has a lot of drawbacks as well, so make sure it works without it first.
This would be a drastic thing to do though just to keep a looping header :), so maybe there is a way to pass some variables from one page to another in flash, so that the animation starts at the correct position everytime - thus creating a loop. Google?
Hope this somehow helps. :)