Find the minimal subset of rows of some matrix such that the sum of each column over this rows exceeds some threshold
The general $n \times m$ case is easily seen to be NP-hard by reduction from set cover. Add a column for each element; each row is the indicator function of a subset; set $t_j = 0$ for each column.

A slightly more complicated reduction from subset sum shows that this is NP-hard even for $m = 2$. Suppose we are given a subset sum problem instance where we have a set $B$ of integers and the goal is to find a subset of $B$ that sums to exactly $k$. We reduce this to your problem as follows: for each integer $b \in B$, add $[b, -b]$ as a row to $A$; then set $t_1 = k - 1$ and $t_2 = -k - 1$. This enforces that the sum of the chosen subset must be greater than or equal to $k$ and also less than or equal to $k$, and therefore exactly equal to $k$.

For solving this problem in practice, I would recommend an integer linear programming solver. Most such solvers have direct support for 0-1 variables, so the linear optimization version you describe in the question can be presented to the solver exactly as written.