largest bitwise-or subset with a maximum number of on bits?
This problem is NP-hard. It is called "the minimum-k-union problem". Given a universe $U$ and a family of sets $\mathcal{F}$ over this universe. Are there $k$ distinct sets in $F$ such that the union of all these sets has size at most $d$.

Your presentation is the matrix formulation of this problem where each column corresponds to an element in the universe and each row corresponds to a set in the family. We are looking for $k$ rows such that the number of columns having at least one "1" is at most $d$.

The Problem is NP-hard and has approximation algorithms. Here is a link for one of them.