2007 macbook pro won't boot. What's the easiest way to diagnose the problem?
The mid 2007 MacBook Pros (MBPro 3,1) and the early 2008 models had defective NVidia GPUs that would cause failures similar to yours. The GPUs would fail at various times (6 months in for some, 3-4 years later for others). When that happened, Apple fixed them without charge.

If your MBPro still has its original NVidia GPU, then the problem could very well be that. Unfortunately, Apple's repair extension program for this program ended on Dec 7, 2012.

You might be able to find a replacement logic board on eBay or Craigslist.