This is potentially a very stupid question but I'm going to ask it anyway. With all these huge buildings such as the Abu Dhabi Mosque, where an unbelievable amount of materials such as marble was moved from one side of the earth to the other, is it possible that if we 'shift enough stuff' that we could change the earths centre of gravity, and potentially alter its orbit?
No, these building are still tiny compared to earth's crust mass distribution. One would need to build whole mountain ranges to detect changes in earth gravity field with high precision instruments. And even those wouldn't changed earth orbit measurably because even a mountain range is tiny compared to the mass of the whole earth.
However mountain ranges (and their rock density) do have a measurable influence on earth's gravity and this measured by satellite missions like grace.
PS: And no, it's not a stupid question. But you could submit it to xkcd's "what if?" section ;) .
Just one important note: no amount of "shifting of stuff" will change the Earth's orbit. The centre of mass will have the same orbit, no matter the shifting (unless you "shift" to escape velocity :)). You'll slightly affect the tidal effects that continuously change the Earth's orbit, but those are already quite weak. It will probably have a bigger effect on the Moon than the Earth, I'd guess.
The centre of gravity changes all the time, including "building huge buildings". However, those changes are miniscule - don't forget how much mass there is in the Earth's core, that's quite hard to dislodge. However, gravity changes are important locally (though buildings still don't quite cut it; something like the de-oceanization of the Netherlands might be closer to the mark). For example, one of the things that lead to the development of the theory of continental drift was mapping of India by the British Empire. Their efforts were hindered by the (relatively) huge gravitational anomalies on the subcontinent - different measurement methods gave vastly different distances for the same paths. Good luck making an accurate map :)
Moving plates also affect the rotation of the Earth - yet another example of the conservation of angular momentum. However, more importantly, so do some human mega-structures - most notably, diverting the flows of rivers and building reservoirs for water power plants. Mind you, it's still not something you'll notice, but it's well within our measurement abilities :)
I believe that the only construction project that has had an effect on the Earth's rotation (not orbit), was the Three Gorges Dam in China.
Even then, the tiny fraction of a second (0.6 microseconds, in fact) difference was because of the mass of water gathered behind it rather than the dam itself.
Also, the difference in rotation is only known by calculation, as it is much too small to be directly measured.
Additionally, please note that powerful earthquakes have a larger effect on the earth's rotation.