Generally speaking, and I'm only speaking for Intel chipsets, when you have a dedicated GPU installed on your motherboard, it automatically disables the onboard graphics from the CPU by default. You can go into BIOS and override the on board graphics option to "always on" instead of off or auto (auto is usually the default setting).
Most people who do this run a second monitor off their on board graphics for web surfing while gaming or whatever. For gaming, you will need to have that monitor plugged into your GPU since it is in no way tied to the motherboard's onboard graphics HDMI output. They are two completely separate video output devices and cannot be used interchangeably for a single output. It's one or the other as your...