Traditionally, most apps behave by storing the file in their own space. This can be accessed later. Which is why most people have to remember to go through various apps (especially stuff like video apps) to delete the information that's saved.
That's what Tim is referring to.
I downloaded MoviePlayer a while ago and so I just re-installed it. Huh. No, it doesn't save e-mailed videos. In fact, I couldn't even make the video re-open (from e-mail ... after it opened once)!
So, I'm going say that I don't know how MoviePlayer works - and recommend you get an app like OPlayer HD ($4.99, USD). There is a Lite version (read: free) that you can try out. Other than being an excellent player for those random e-mailed videos, OPlayerHD does keep a copy in its files so you can go back later and view them again.
I know this doesn't help in using MoviePlayer - but that's all I've got. Good luck.
Marilyn
P.S. Not a slam against MoviePlayer, BTW. It just doesn't work the way I want ... and I don't know how to make it work like you want.