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First of all, let's agree on one thing:

Stacking cubes together in that pattern (with successive scaling) produces the same shape as digging it out. Try this: dig out one iteration, then multiply in size by three. It's the same as stacking. Or, if you prefer, stack one iteration of cubes on each other, and divide in size by three. They're all equivalent. In fact, it can even be described by an L-System of a line. So it's equally valid to argue it has no surface area. It's 3D cantor dust, really.

Mass=Density*Volume. Volume=0, so Mass=0. Of course, there's the issue where the gaps become smaller than a planck scale, at which point this all becomes meaningless. On the other, at much higher scales, the gaps become small enough to exclude any air molecules - or at least the soot and dust, which is heavier. So the sponge, filled with these small vacuums, might end up floating up until it reaches clean air. Or it could sink, due to the particles (which [maybe, see next sentence] are still there!). It all depends on whether on not it has mass at a quantum level or not.

Define your terms before you start arguing about them. It's like the monk asking, "How does one attain happiness?", and his teacher responds, "What one earth is that?"

You guys! Can't a body dream? Does everything have to be correct in somebodies eyes? It is enough to ask the question and come up with a godzillion of answers! That is the fun! That is my reality

Jeah, I like this too, and was going to put out the fire with gasoline... 

If I get hit by an infinite amount of infinitely sharp edges that are infinitely packed together (are they?) how will that affect me?
 

That's close, Nahee.
The only two differences: The "cords" of meat all have different sizes and are square

Btw, the whole cube would cut equally easy throught the earth, splitting some atoms on its way to the core, lol. (Well, at least assuming, it had any mass.)
If it had no mass, it would basically stay in place where it is and wouldn't ever move from there, but everything that touches it, would be cut into pieces. (Well, additionally assuming, it can't either be affected by any of the other forces^^)