I have a Toshiba Satellite L675. I tried to turn on the laptop and I noticed that the screen would not turn on at all. The light indicator for plug-in power works and the indicator for the computer turns on when I press the power button. The fan spins up normally. The CD drive boots up. The hard drive also spins up (I tested it in another system and it works). I have tried removing the battery and all power and then holding the power button down for over 60 seconds with no avail. I have tried to connect the laptop to another monitor with no avail. I have removed and re-inserted the ram too. I notice that the hard-disk access light on the laptop does not flicker at all.
The symptoms you have mentioned and the troubleshooting you have tried so far, indicate either of the two possibilities:
1. The Video chip on the motherboard or the video port is non-functional.
2. A rare possibility that both the Motherboard's video controller and LCD have gone bad.
As you mentioned that you tried to connect the Hard Drive to a different machine and it worked, if the other machine is of the similar make try to connect the LCD from this machine to it and check if it works. However, this is a complicated step and needs technical supervision. You can refer to the service manual for steps on how to remove the LCD and for safety instructions.
If you get video by connecting the LCD to another computer then it indicates the problem is with the motherboard or the video port and needs replacement.