What happens when we increase or decrease capacities in the minimum flow?
Case 1, increase the weight.  Consider a path on at least three vertices with capacity 1 for every edge.  Max-flow is 1, min-cut is 1.  Increase one of the edges' weight, and the max-flow will still be 1, hence the claim is false.
Case 2, let $C$ be a minimum cut with cost $w(C)$.  Decrease the weight of one of the edges in $C$ to get a new minimum cost $w'(C)$.  Since $C$ is still a cut in the graph, but $w'(C)$ is smaller than any other minimum cut, the claim is true.