I don't know why you think that misting caused the leaf edges to turn brown. Did you mist with something other than water?

Brown leaf edges are more liklel a symptom of improper watering. If your Dieffenbachia is in the right sized pot, then water thoroughly as soon as the top half inch of soil feels dry. If your tap water is hard, use filtered, distilled, or rainwater.

Trim off the damaged edges. If you correct the watering problem, the plant will probably forgive you.