What does the tamper button on my STM3220G-EVAL board do?
From a quick view into the datasheet, the "anti-tempering" is advertised as a feature of the microcontroller. The chip has hardware support for anti-tempering measures and the button is on the dev board for you to test and evaluate these features. See http://en.wikipedia.org/wiki/STM32#STM32_F2:


  Static RAM consists of 64 / 96 / 128 KB general purpose, 4 KB battery-backed, 80 bytes battery-backed with tamper-detection erase.


You could use this feature to prevent sensitive data from being read from the device by opening the case and hijacking the battery-backed memory.