Nothing would happen. Dish washing liquid is most likely non-toxic. If it gets in your cut I don't see it causing anything bad - if anything it will be good ..They use Dawn Dish Detergent to clean them ducks who come from oceans that survived oil spoils and are covered with thick black oil and once they've washed those ducks then they send them to place where they go on on living a good productive life. I've always used dish detergent when I got cut / wound. Never hurt me yet.
