Pressure release and earthquake
Let's say the earth is flat. Someone standing at the center would feel weightless. The closer he gets to the edges the more matter he will have behind him and the stronger gravitational pull will be. He will literary feel like he is climbing a steeper and steeper mountain. It's the same for the matter which composes the Earth. Think of it as a sponge. As soon as you release it it will expand and take more place. It the same for matter: at the center it weights nothing but it is squeezed between the matter which stands at the edges, willing to collapse. This squeezing (pressure) compresses matter which consequently will to expand as soon as possible. The moment you suppress gravity you allow matter to expand and release its potential energy as a sound wave travelling through Earth.
In my opinion if you target a block which is inside the crust this would look very much like an earthquake and would have similar impacts on the environment. If you activate this on a volume that contains crust and air I think that everything described by <PERSON> is correct.
No centripetal force anymore
According to <PERSON>'s laws of motion, any object to which is applied a force $F$ will move through empty space in straight line forever. It's linear motion and no force is required to actually move. 319
Force is required to change the momentum (direction and speed). Objects on Earth's surface are not moving in straight line because of a centripetal force which keeps changing their momentum. They want to move in linear motion but they can't. They are like a rotating ball attached by a string. This force is Gravity.
The Earth is spinning very fast. The angular speed of Earth's rotation in inertial space is about $7.2921150\times 10^{-5}\ \text{rad/s}$ (solar second). When you multiply this by equatorial radius of $6\,378\,137\ \text{m}$ it gives you a tangential speed of $465.1\ \text{m/s}$. If you were able to remove gravity from a block of matter, nothing (as far as I know) would stop it from moving in straight line at $465.1\ \text{m/s}$ into space except for the frictional forces and the effect area. Friction might be quickly overwhelmed as your block increases in size.
Scratching the surface
I meant to have the area of effects not penetrate the surface on any of the scenarios except the last one
If your machine does not target Earth's crust at all things tend to be gentler. Imagine everything <PERSON> talked about without any violent explosions. On a large scale however, air displacements would significantly affect the weather, producing exceedingly violent winds travelling at huge speed and destroying almost everything on its passage.
The speed of light
If your matter losses its mass it must become something massless and there is not many options here: it should turn into energy (otherwise your Universe has energy leaks which is very weird and question all modern physics). According to the mass-energy equivalence equation $E=mc^2$ this is completely devastating and equivalent to an anti-matter bomb. Since I suppose this is not really interesting for your story, you have to take this into account, either to ignore it of find a way to avoid it.
How do you do this?
The how is not important, I want to know what would happen after it was turned on at different levels.
I disagree and how is certainly a question you should ask to yourself and be clear about in your question: how does your machine remove gravity from the block of matter? Does it modify space-time curvature? Does it make you matter massless so almost unaffected by gravity (turning it into electromagnetic waves and therefore making it travelling at light speed)? It might impacts the answers you obtain here. 319