Perhaps your office mate meant it's the "preferred" or "more common" way; neither is incorrect in American or British English.
I can think of two points that may help you decide which you'd prefer to use:
First, the phrase "this computer" can be removed entirely (since the meaning of the sign would be obvious in context). Doing so would make the phrases identical, with "turn" and "off" consecutive words.
Second, reversing the composition becomes "Please leave this computer on." The other phrasing, "Please leave on this computer," is awkward.
As an editor, I would go with "Please leave on" or "Please do not turn off" depending on your penchant for brevity. This question really comes down to personal preferences.