Can the null space for a matrix with more rows than columns be restricted to 0 dimensions?

A matrix has a $0$-dimensional null-space if and only if its columns are linearly independent. This follows immediately from the definitions of null-space and linear independence.

If the matrix is square, then this is equivalent to its columns being linearly independent (and to the matrix having maximal rank, and a non-zero determinant, etc.)

If the matrix has more rows than columns, then its rows cannot possibly be linearly independent, and so linear independence of the rows tells you nothing about anything.

If the matrix has more columns than rows, then its rows may or may not be linearly independent, but the matrix cannot possibly have a non-zero null-space.