The zener diode, D3, is a crowbar. It is supposed to draw lots of current should the voltage rail exceed 5.6V. It will either blow any external fuse or give its life keeping the VCC (5V) rail from going too high. Be careful when purchasing a replacement. The voltage rating of the diode must be somewhat higher than the expected voltage. You cannot, for example, substitute a 5.1V zener.
I would suggest replacing D3 and then with UC1 removed, power the unit from USB and measure the voltage at pin 5 of IC1. This is the feedback voltage to the switching regulator. It must be at 1.25V. If this voltage is wrong, check resistors RT and RB.
Next, make sure that diodes D1 and D2 are not interchanged. These diodes have very different characteristics. Also make sure D2 is installed with the correct orientation.
Finally, replace IC1 and with USB disconnected apply 24V AC. Quickly check VIN (5V) and +3.3V, then power down and feel around for anything getting hot. If voltages are normal and everything is cool to the touch. Then give it a longer power up cycle. If still OK, then populate the rest of the ICs and LCD and do a full test.
Personally, I would not recommend plugging in 24 AC and the USB cable at the same time. Doing this connects the VIN of the OpenSprinkler with the +5V rail of the computer via the USB port. This may not be an issue on some computers. However, it could problems on others. This is because you could be tieing together the outputs of two different switching power supplies. Again, if the computer is well designed, it may not be an issue. The computer is supposed to have these ports current limited. The downstream device, here the OpenSprinkler, is supposed to be current limited too.