How to Repair 15" Mac Book Pro Core 2 Duo 2.16 Ghz That Won't Power On?
In the absence of “chime” there are three things to blame: RAM, GPU or Board/CPU. 

I know because I had the same problem throughout the years with different Macs (and even Powerbooks). Try removing some RAM and putting some other modules, also remove the battery as suggested by Philip in the comments. 

Using an external display may help identify screen problems too.

And finally an Apple Store, even if it’s out of warranty, will diagnose the box (sometimes for free, it depends) sometimes for a small fee. Bear in mind they will always go for the brute force route: “you have to replace X,Y,Z”, 1000 u$s. Thanks for Coming, Bye. ;)