I am running elementary OS on a laptop with Intel Haswell integrated graphics and I can no longer detect my external monitor via HDMI. It was working perfectly for the past month, but suddenly today it has stopped working. I don't remember doing anything significant to affect it.
Have you recently installed updates?
It's always possible there was an issue / conflict in a new package. Installing new updates could help. You could try booting up to live USB and see if the HDMI works in that setup? Then at least you'll know it's not physically broken hardware or cable.
You can use AppCentre or on the command line type:
sudo apt-get update
sudo apt-get full-upgrade
See if this helps, otherwise checking with live USB is option.
If you have added any additional PPAs they could also have produced a package conflict. Hope this helps.