As of right now, the late 2006 (which I believe is the "first run" of the "Intel iMacs") -will not- run the upcoming Mountain Lion OS.
I believe the model you're talking about is referred to as the "revision B" of the white Intel iMacs. I have a white Intel iMac 24" (purchased in April 2007, just after the first "aluminum" iMacs were released), and it will not run Mountain Lion.
The 24" white iMac can run ML. Looking at the specs published here on MacRumors (https://www.macrumors.com/2012/02/1...many-2006-2008-macs-with-integrated-graphics/), the line with listings for iMacs states "Late 2006 polycarbonate iMac (Model Number: MA710xx/A)" as being not compatible with ML, but compatible with Lion. The 24" white iMac is not model number MA710xx/A, but MA456xx/A. There's is also a line stating "Systems using ATI's Radeon X1600 graphics will also be incompatible with OS X Mountain Lion.". This rules out the 17" and 20" late 2006 white iMac, but not the 24" which uses the NVIDIA 7300GT/7600GT.