The following article details a number of things worth checking, just in case they haven't been checked previously, that may help rid you of the little black box.
I would suggest logging a call with AppleCare as Apple don't seem to have too much visibility of the issue.
I have the issue as well. I find that if Menu out of the Film, the go back in and resume the issue goes away for the remainder of the film. If Start from the beginning the black rectangle in subtitle place appears and does not go away.
If you turn the brightness up then you should well see that the black box is transparent and you can see through to the movie underneath which is playing.
Hopefully if enough people raise calls with AppleCare then it may well get fixed,as the AppleCare advisor has said that this isn't an issue that they are aware of.
I can confirm that the issue went away with an AppleTV Mk3 that I borrowed from a friend.
I am convinced that is an issue with subtitling.
The iTunes Store support just seem to offer refunds or recommend you contact AppleCare to resolve a technical issue with your AppleTV or your library.
I have completely rebuilt and redownloaded the mini my library is on. Also tried a different account and the content has been replaced in my account but makes no difference.
Stop move and then restart movie and you get the ...."from where you left off" or "from beginning" option....chose, "from where you left off" and guess what, no black matchbox.
Something must be wrong somewhere, but this seems to work, not great though having to do this every movie.
Hope it works for you.