Z80 Simple Computer Troubleshooting: IORQ pin not going low
Yes, the /IORQ pin should go low when an OUT instruction is executed. The fact that yours isn't suggests that the program is crashing or not running at all.   

There could be many reasons for this malfunction, such as a wiring error, faulty chip etc. I would start by checking to see if the CPU can work by itself. To do that I would:-


Remove the VIA and EEPROM. Check all wiring for shorts etc, and make sure the power supply and bypass capacitors are connected to the correct pins. 
Tie all 8 data lines to ground via resistors (~1k each), to create a NOP instruction.
Power up the circuit and check that all input signals are correct (/INT, /NMI, /BUSRQ, /WAIT, /RESET should all be high)
Reset the CPU, then let it run and check for activity on /M1, /RFSH, /MREQ and /RD.  


Assuming that test passes, change the opcode to OUT(#D3),A by raising all data lines to +5V except D5, D3 and D2. Then restart the CPU and look for activity on /IORQ and /WR.

If that doesn't work then the CPU is probably faulty (or a fake!). 

If those tests pass then remove the resistors and put the EEPROM back in. You should get similar results. If you get activity on /M1, /MREQ and /RD (indicating that instructions are being read and executed) but nothing on /IORQ, check that the data and address lines are matched up between the CPU and ROM, and that the ROM is putting the correct data onto the bus.

If that works, then there must be problem in the VIA or its wiring.