I know my response is a few years late but I had the same thing happen to me and my computer is working now. My story might not help since I never found the problem but here is what happened to me...
I had just finished a long editing session on my Mac Pro workstation and shut the computer down for the night. In the morning I turned it back on and and was sitting at the white screen of death, never making it to the apple logo. It's rare but it happens sometimes so I manually powered down.
- First I tried resetting the PRAM with Option-Command-P-R on startup. The first chime happened and then the second chime only partially happened and then kept chirping over and over like in this video posted about.... http://www.youtube.com/watch?v=tQuo-67iC... The only difference with my situation was the chirping chimes were about half of the duration. Probably about 100ms long chimes. I tried restarting after without the PRAM reset command and the same thing happened with the short chimes repeating.
- I tried SHIFT on startup but Safe Boot mode didn't work.
- I decided to try booting from the Restore DVD. The optical drive tray wouldn't open with Eject, F12, or holding the mouse button down so I had to pull it out of the bay and manually press the button to eject the tray which worked fine. I held C on startup but just ended up at the white screen again with the same repeating chime. Then I tried starting while holding option to see if I could boot from the Restore DVD volume via Startup Manager. That didn't work either. Same result.
- I removed the 32GB of ram I'd upgraded to and replaced it with the measly 2GB of stock Apple ram that it came with. Same results.
- I removed my upgraded 2GB GTX 680 graphics card and replaced it with the stock 512MB ATI 2600 XT card and that did nothing.
- I tried removing all hard drives except the boot drive. Tried removing all hard drives including the boot drive. Tried removing the optical drive in combination with only the boot drive in and with no drives or optical drive. Lastly I tried booting off the Restore DVD with only the optical drive plugged in. Nothing worked.
- The whole time I only had one Apple Cinema Display plugged in and only the Apple keyboard and mouse that came with the computer. I kept the devices connected to the computer to the bare minimum the whole time while troubleshooting to make sure there was nothing plugged into any ports that could be causing conflict. I also blasted what dust I could see with canned air when I started just in case.
- The very last thing I did was try unplugging the keyboard and mouse and starting the computer and that didn't work.
That's when I gave up, figured it was a logic board issue, and started scraping up funds to get a replacement workstation asap.
PROBLEM SOLVED >>> I spent the next two days of my life hating life because I had gone from working on my super fast 8-Core Mac Pro to working on my 1.3GHz MacBook Air. My work is pretty demanding of the CPU and GPU and operations on the laptop were taking between 4-10x as long depending on what software I was in. I literally said to my computer "Why can't you just turn on?" and in a moment of desperation I decided to press the power button one last time. My computer started up completely normal and was booted into OS X in a matter of seconds.
So it fixed itself somehow. It sat idle for 2 days and whatever happened in that time worked itself out and it magically turned on. I know this write up probably can't help anybody in the same situation get back up and running since I don't actually know what the solution was. It does drive home the fact that that this is indeed a mysterious problem.
POST FIX - After it turned back on I shut down and booted up a half dozen times with my fingers crossed and everything booted like normal. Then I started putting the pieces back together one component at a time and rebooting each time to see if one of the components caused problem to return.
I added the the 32GB of ram back and rebooted and everything was fine. Then I added my internal hard drives back one by one rebooting each time. Then the optical drive got plugged in with a normal reboot. Last came swapping out the stock graphics card with my upgraded GPU. And there it is. My computer was back to it's fully upgraded self and running like a champ.
The problem and the fact that it fixed itself after just sitting there for a couple days is beyond me. It's not heat related because the problem started after it was off overnight so it started cool. I can't wrap my head around it.