No, the header can only be a static image (no video or Flash files).
On an additional note, the Console doesn't have an embedded player for audio and video files. Still, you can embed video players and files hosted on 3rd party services like YouTube, Vimeo, AudioAcrobat, etc., as long as the 3rd party can provide secured URLs (of the "https" type) to the files hosted there. Otherwise, the Console being a secured website, the browsers will not allow the display of insecure content (the "http" type of file links). YouTube, Vimeo and AudioAcrobat, for example, provide such secured ("https") links to the hosted audio and video files.