I dont think it need a monitor to boot.
However if no monitor is connected/identified during boot the graphic will be disabled (which is a common thing) to save power.
Now the problem comes when you run a GUI based OS such as windows, because according to windows there is now suddently no graphiccard to draw the GUI at so it will fail.
Compared to lets say a Linuxserver who doesnt need any graphiccard to handle incoming SSH-connections.
There are hdmi-plugs one can buy and connect to the output which will fake a monitor so the graphics driver will be happy and the OS will be happy to have a graphic card to draw its GUI onto (and suddently you can now login through RDP to your Windows box).
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I can confirm that the DN2820FYKH can boot fine without a monitor.
There have been issues with some BIOS versions; I am using BIOS 47, dated 24.12.2014.
Operating system is now SparkyLinux with Openbox, Version 3.6, with EFI boot. Setting up EFI boot was on-trivial, but possible; I can post some hints if anyone is interested. Note however, that I had to tick the "Legacy Boot" box as well as the UEFI Boot box to get the NUC to boot without the monitor. However, I have verified that it is indeed booting in EFI mode.
I had previously checked that headless boot was possible with Ubuntu Server 14.04, with the same BIOS. It was not possible when I had BIOS 40.