The greatest role of the supernatural in Shakespeare's play A Midsummer Night's Dream is the setting itself (which includes the location of the play and the characters who inhabit the dream-like world). The setting of the play includes a fantastical world filled with spirits and fairies. On top of these characters, dreams and spells also add to the supernatural collective in the play.
The play, overall, exists in a world created to be imaginative. Through this imagination, both humor and love are attended to in a way that only the supernatural can breed. Characters are changed, love is tested, and spirits fly about in order to bring chaos to an already chaotic world. Without the inclusion of the supernatural, the play would have the substance it needs to teach the lessons embedded in the action.
Essentially, without the supernatural, the play would lose its magical setting, the characters, and the lesson. It is only through the continuous use of the fantastical that the play can reach its desired climax.