I know this is not a common scenario, but for the few people that upgraded a Mac originally equipped with a Core Duo processor to a Core 2 Duo processor, here is the procedure. I have an iMac version 4,1 that I upgraded years ago, and it worked. The person who found the solution had an upgraded Mac Mini. You will need a second computer that is Lion capable for this tip to work.
As always, Make sure you have a backup of the original install! This worked for me, but always be prepared for the worst. 
- If you attempt to install or update the HP software on a Mac running 10.8 Mountain Lion using the software CD that came with your printer, or using a previously downloaded driver, an ' Unsupported Operating System: This installer is not compatible. ' or ' This HP software may not be compatible with your version of Mac OS X ' message displays on the computer.
- Install Mountain Lion On Unsupported Mac; How To Install Mountain Lion On An Unsupported Macbook; Apple have always been impressing the crowd with their software and hardware enhancements over the year. To add a feather to that cap, Apple has redefined the operating system experience with the Mac OS X Mountain Lion 10.8.
Mountain Lion Patch
Installing Mountain Lion on an unsupported Mac involves defeating these issues. Where there is a challenge, it is a safe bet that hardworking hackers will rise to the occasion. Perhaps you should have an Apple tech check out your system instead of complaining about the problem here.
Amd A10 Hackintosh
I made, and used, a Lion Mac OS X Install ESD on a thumb drive. Instructions to make one can be found all over the Mac centric web sites. Then follow this procedure:
- Put the unsupported Mac in Target Disk mode and plug it into a Lion supported Mac.
- Run the Lion installer and make sure to choose your unsupported Mac's hard drive. Let the install finish.
- Boot back into the supported Mac's OS, and put the unsupported Mac back to target disk mode.
- Delete the PlatformSupport.plist at /System/Library/CoreServices on the unsupported Mac's hard drive.
- Reboot the Unsupported Mac and it should boot into Lion normally.
[crarko adds: