On a i7-720QM or i7-820QM mobile processors, can each core access all of the available RAM (i.e. 8 GB out of 8 GB). I've heard that on certain desktop motherboards this can cause problems, but I can't find a source.
Each core can access all the memory. I'm not aware of any desktop/laptop/server that any system has that limitation.
Perhaps you are thinking of multi-socket Athlon systems where the RAM is connected to each socket, but even then each core could see all the RAM, some was just slower to access (that's called NUMA if you are interested)
Or maybe you're thinking of the 4GB limitation for 32 bit Windows XP/Vista/7? You will need a 64bit version of Windows to access all that RAM.