Interesting, I just e-mailed BFG support last week about the same exact problem.
I've played this game through once before, went to play it last week, and it "crashed". It actually just did not start after selecting the first book and then selecting "start case". It closed and went back to the game manager desktop, and stated "Thank you for playing The Lost Cases of Sherlock Holmes" as if my time ran out on the demo.
I've gone through all of the typical trouble shooting, and my computer has no problems. I installed it on my brand new laptop, and the same thing happens. I'm thinking it has something to do with the demo time limit script being active. Kind of weird since I did buy the game.
They have not figured it out yet, but I will post here if they come up with a solution.