Are you using any kind of adapters to connect the monitor or is it straight HDMI to HDMI connection? I had troubles with non-apple adapters – the simply dind't worked for me...
You can make 2 things to diagnose the issue I can think of:
boot into linux (make a usb from ubuntu with unetbootin)and check if it works in linux – if it does it's a software issue (i guess), meaning some kind of hack is necessary
system.log by opening the Console.app or by tailing it in terminal with
tail -f /var/log/system.log, also take a look if
sudo dmesg got anything useful to say.
When you got the console or terminal open – just connect/disconnect the monitor and check if you get any kind of log activity from it, that should at least give you some idea if the hardware reacts on the display when Intel HDxxxx is used for graphics.
I would first check the syslog and then check the setup in Ubuntu 15.10, grab the latest Ubuntu iso file from here: http://cdimage.ubuntu.com/daily-live/current/wily-desktop-amd64.iso