For a given sparse matrix, how can I multiply it with a given vector of binary values
The main problem is the size of your problem and the fact you're using Python which is on the order of 10-100x slower for matrix multiplication than some other languages. Unless you use something like Cython I don't see you getting an improvement.
I have a sparse matrix and another vector and I want to multiply the matrix and vector so that each column of the vector where it's equal to zero it'll zero the entire column of the sparse matrix. 
How can I achieve that?