monotone min 3-sat polynomial algorithm?
The problem MIN-3SAT is usually know as "minimum vertex cover for 3-uniform hypergraphs". Any reasonable algorithm that you will suggest will give a 3-approximation (or worse), and indeed, it is conjectured (and follows from the unique games conjecture) that no polynomial time algorithm can improve on this approximation ratio.

This means that if you try hard enough, it is extremely likely that you will find an instance in which your greedy approach gives a solution which is roughly 3 times as worse as the optimal solution. I'll let you find it on your own.