Why do you want to remove the icon? Is it to stop people from being able to download the video, because that won't work - anyone with the slightest technical knowledge of browsers will still be able to get the URL of the video and download it directly.
If it is just to tidy up the interface a bit, then, apparently, changing the output HTML for the video player to have:
as an attribute will remove it. I've not looked to see if this can be done without changing core code.