Intel Motherboards : Fatal1ty z170 gaming-itx/ac 6600k hdmi does

Firstly it is worth noting that the motherboard does not possess a built in PC speaker so unless you have a PC speaker in your case hooked up you will not hear a post beep. 

Based on your symptoms as you described the system powers on but nothing is displayed on the monitor. This could be caused by a broad variety of issues. It is good that you tested the monitor and cable on another system, that at least rules out a defective monitor. Also based on your description the system does not instantly power off when you press the power button. This means the system is likely not posting.

If you have a PC speaker in your case you should connect it to the "Speaker" header on the motherboard. If you do not have one in your case then you can pick one up cheap, this one on newegg is a little pricey for what it is but shows what I am talking about.

With a PC speaker installed you will get more feedback from the system letting you know what is wrong.

For now though:

Check power connections are connected (both 24pin and 8pin CPU power)

Be sure the 8pin is actually the 8pin for the motherboard not an 8pin PCIe power connector

Install only a single module of RAM

Clear CMOS, preferably via this method

Disconnect all SATA cables, drives, M.2/mini PCIe devices etc from the board

Power the system and see if you can get POST.