Shift registers(?) failing after being left on a while
Your issue is that the SRCLR input to the 74HC595 IC is disconnected. Believe it or not, I actually had this exact same issue on an earlier circuit that I designed. What happens is that the pin will pick up random signals and clear that shift register, which subsequently clears the display. You should tie that input to high.

The other thing is that you have no resistors in your design. While the 595's do have a maximum drive capability, which keeps the LEDs safe from destruction, you should nonetheless add some sort of resistance there. It might damage the LEDs or damage the 595's.