Is wiring with melted insulation and exposed copper dangerous?
Sounds like a fire waiting to happen. You'll need to cut out the damaged part and splice them together again.

If there isn't enough slack then you'll need a new short run of wire to bridge the gap.

Because you have physical access to the area The splices should be inside junction boxes.

Bill the work/materials to the plumber or deduct it from his bill if you haven't payed him yet and tell him to invest in a scorch pad to avoid stuff like this.