Yes, if you take out the CPU you can install it on another motherboard as long as the motherboard is compatible. You don't have to worry about Intel CPU's fitting only into HP, DELL etc. motherboards; an Intel CPU will fit into any compatible motherboard which supports the processor.
If you do purchase the computer for $120 or even be it £120 (which I still think is a good deal), you can keep the motherboard and simply mount it into a bigger case. You don't need a new motherboard for installing larger components, such as a GPU; you only need a bigger case. Of course if you want to change the motherboard, that's still no issue either.
To answer your question simply, yes you can take out the CPU and install it on another motherboard. Just ensure the other motherboard is compatible, for example if it's a i7-2600, make sure the motherboard you want to install it into is on socket 1155, and is a 6 or 7 series motherboard, e.g. H67, Z77 etc. If you need more assistance on what motherboard the i7 is compatible with, please reply back with which i7 it is, example i7-3770 etc. and I'll state which socket/chipsets are compatible; or create a new thread and we'll assist.
All the best.