The problem with the compactified dimensions described in that book is that they're way too tiny. Greene's extra dimensions are as small as one Planck length, and the macroscopic objects are as large as one Planck length in thickness in each dimension. So, you can't hide anything by rotating it through the other dimensions, any more than you can hide a 1cm-thick rope by rotating it around its axis.
However, with the later ADD1 theory, where the fundamental scale is as low as the weak scale, some of the compactified dimensions are more on the order of 1mm, while the thickness of most objects is still down near the Planck scale.
This is an easy way to get braneworld physics. For example, with 3+1 normal dimensions, 2 large extra dimensions around 1mm, and 4 tiny extra dimensions around the Planck length, you can have multiple 3+1 universes constrained to 3+1 branes, that are almost entirely independent of each other, but still interact via gravity—which falls off as
r^4 instead of
r^2 across the compact dimensions, making a difference small enough to not invalidate all of our previous experiments2 but just large enough to make a difference.
In that case, rotating an object through one of the large extra dimensions would essentially mean moving into the empty space between branes in the multiverse. A tiny rotation (even near the Planck scale) would be enough to make an object invisible to electromagnetism (and the strong and weak forces). But, with sensitive enough detectors, it could still see our universe's very pale gravitational shadow, and vice-versa.
And meanwhile, if our brane is sandwiched between two other branes, rotating could mean making contact with a whole other universe, possibly filled with weird and dangerous aliens, who maybe even live according to slightly different laws of physics.
Which opens the door to all kinds of spare-opera plot devices.
The big problem is how you get the force to push or pull an object on those axes. EM, weak, and strong forces can't propagate in those directions. Gravity can, but any reasonable gravity (or anti-grav magic) would be likely to push you much harder along the brane than it would off-brane. Your best bet there would be to not describe how it works. Keep your pseudoscience technobabble less specific than Star Trek. Maybe mention helicity or twist and leave it at that.3
As for what it would be like for people watching objects pushed out of our universe—they just vanish. One microsecond they're reflecting light, the next they aren't. And the same goes for people inside spaceships or other objects that get pushed out of our universe—all of the light from stars and galaxies, or reflected off close objects, just disappears instantly.4
1. Named for its inventors, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali.
2. Unfortunately, LHC experiments and neutron star observations done since then to measure mm-scale gravitational effects seem to rule out the original ADD model. Of course you can adjust things and come up with a new theory that's still compatible with all of the results, but many of the original motivations for ADD are no longer so compelling, so not much work is being done there.
3. Around the turn of the century, it was popular to throw in the term Calabi-Yau manifold, and most readers would just assume you know more physics than them, instead of less. I'm not sure if that still works a couple decades in; maybe better to just say "superstring" or "D-brane".
4. At least far faster than a human could possibly detect. If you have super-fast AIs and want to try to describe their perceptions, maybe they see it quickly fade out. But they're probably also directly experiencing those "gravity shadow sensors", right? So, they see gravity shadows receding at r^4 instead of r^2, like everything in the universe is becoming quadratically more distant, which is probably a more interesting distortion t describe.