Your GPU will turn on, fans spin, there will be LEDs for the PCIe power cables if they are not connected. As for damage, I'm not qualified to answer that part, but it happened multiple times. Card still ran.
In my previous experience, when I powered up my system without the external power connectors, my system gave a high-pitched screech which continued until power was shut off (ATI X1800XL). More recently, I've read that systems simply do not power up without the proper power connections to the graphics card. My assumption is that they fail POST.