Q:

What do I get by multiplying a 0 operator on a 0 eigenvector?

I don't know how to write the equation form. Assuming my notation as Dirac notation, what do I get from 
$$ ( 0 | 0 | 0 ) ~?$$

A:

It is not quite clear what you mean with "0 eigenvector". However, acting with the "0 operator" on any vector gives zero (i.e., the zero vector). So the answer to your question is definitely "zero",
$$ \langle \psi | 0 | \psi \rangle = 0 . $$

