Depends on the GPU. I do not have a MSI R9 for testing. Even if I did, I wouldn't unplug the power connectors for your question.
I know that Nvidia Quadro series (K2000 thru to the M10 tesla cards) do - they just won't let Windows know if there's a monitor attached; but it shows in device manager (assuming you can access via another display or RDP). If you run Nvidia's Control panel or the CLI tools it'll actually prompt you that the card is not properly powered.
It is logical that the card behave in unexpected ways. It may not detect at all, it may detect as device which is expecting a firmware update, or it might detect as a display adapter but just not output a picture. To answer your question in 1 line: yes, it is a possibility that your PSU connectors aren't working properly and that's why the GPU isn't working properly.
The internet does have reports of cards breaking themselves by not plugging in the correct power so exercise caution. Personally I'm doubtful these reports are real.