I did some research and found out that this is rather an Apple issue. Here are some useful articles and forum threads that I found:
I don't this is a Sitefinity issue. I also found this article - it looks like the video itself must be in a specific format so the iPhone/iPad plays it
I did some research and I found the following projects that implement X-Sendfile functionality for IIS:
I hope they will be useful for you.
I have logged a new PITS feature request for this - here is the public URL
You can vote for it and if it gets enough votes it will be implemented.