Hi Sven
TinyDB is saved in a private folder dedicated to the App. The casual User cannot access it. However, you may be able to get to it using a 3rd party file explorer.
The snag is, you are going to have to try to access the tablet with another device, given that the tablet screen is dead. So I suggest connecting your tablet to a Windows PC/Tablet and try Droid Explorer:
http://de.bit13.com
I do not know if you will be able to dig-down to the TinyDB file, but it's worth a try.