This is an opinion question, so I'm giving you an opinion answer of how I would do it. There are many, many other ways both on the shader side and the mesh side.

The idea: use a shader that reads mesh vertex color (http://docs.unity3d.com/Documentation/ScriptReference/Mesh-colors.html). There are different ways to do this depending on whether your shader is written with ShaderLab, vertex/fragment (Cg/HLSL), or as a surface shader. How to do this is a separate question, but here's a couple links to get you started:

• ShaderLab binding of vertex color for use in the shader: http://docs.unity3d.com/Documentation/Components/SL-BindChannels.html
• Cg shaders: http://answers.unity3d.com/questions/189584/how-to-get-vertex-color-in-a-cg-shader.html

Whenever you need to calculate which vertices of the mesh are visible and which are not (every frame while your sword is getting pulled), you loop through the vertices and assign the vertex color like this pseudocode:

foreach (vertex in the mesh)
{
  if (vertex is in the bounding space you want to be invisible)
    vertex color = white
  else
    vertex color = clear
}

Now all you have to do in your shader is multiply the color by the vertex color. Multiplying by white won't change the color, while multiplying by clear will result in clear. You could even do fancy effects like energy fade-ins: if inside the bounding area, make clear, if just outside the bounding area, fade from clear to bright green (or whatever color), if further out fade from bright green to white.

For your last option (loss of limb), you'd have to do some procedural mesh generation that's worth a separate question and a bit of math. Rather than changing the vertex color of vertices in the bounding area, you'd remove them from the mesh entirely, cut their triangles with the plane of the bounding area, build a ring of the cut triangles, and create a mesh around the ring.