Probably the sensors on the control board are not making contact with the buttons. I have fixed this before on a few PS3s. Follow this guide: PlayStation 3 Control Board Replacement
On the very last step (step 9) you can see the sensors on the left and right of the control board they look like thin metal buttons.
- Once you have the cover off and the control board exposed, plug in your PS3 and attempt to turn on the ps3 slightly touching the right sensor with your finger.
- DO NOT PUSH DOWN HARD ON IT!!! all it takes is a slight touch.
- If the control board is working then the PS3 should turn on. If it does not then you can inspect the ribbon cable to make sure it is connected properly. If it still does not work then your control board is probably dead.
- Once you have determined that the control board is working then use a pocket knife or a credit card or something thin to pull up the thin metal sensors so that they make contact with the buttons. BE CAREFULL NOT TO PULL TO HARD OR THEY WILL BREAK OFF.
- Put the cover back on and make sure it now turns on using the buttons DO THIS BEFORE YOU PUT THE SCREWS BACK ON.
I have done this a few times and it has worked every time. If your LED lights still come on then your control board is probably ok and just needs to pull the sensors up just a bit. It takes about 15 to 30 minutes to do this.
I hope this helps, good luck!!