Hello, same problem here. First I thought I had a defective TFT, but also a H-bridge or any other components won't work. To get it work properly I have do disconnect and reconnect the power supply, or open the serial monitor in the arduino IDE. Also the reset-button on my board doesn't seem to work.
I am using IDE 1.5.7 and a Arduino DUE board from ebay/china (about 16€).
It would be great to find a solution because I need the DUE to start correctly in my project.
I hope someone knows the reason for this behaviour.