To me, dreams are important in the play, but I do not think they actually affect the outcome. Instead, I would say that the dreams emphasize the meaning of play.
The whole play is meant to be something of a fantasy. There are fairies and love potions and mistaken identities and stuff like that. Because of that, the characters are often unsure if they are awake or dreaming. The dreaminess of the plot line, in my opinion, emphasizes how unreal our lives can feel -- how strange and out of our control they can be.
But the dreams don't really affect the outcome. When Hermia dreams of the snake, for example, it doesn't really change what she does next.
The importance of dreams in William Shakespeare's play A Midsummer Night's Dream lies in the ancient and traditional countryside customs of history.In Shakespeare's time it was common in England to try to divine the future of loves and lives at Midsummer - through dreams and other superstitions. One could have one's fortune told about the likelihood of a relationship lasting, the likely loyalty of a new young love and so on. There was no such season as Spring in Shakespeare's time, only Autumn, Winter and Summer - so the play does in effect take place at their Midsummer. The start and height of summer was a big deal, with many festivities such as May Day and Dawn services. Young maidens also used flowers to guess their romantic fortunes - dreams however are fleeting and fickle and not to be relied upon, like the events in the play.
Shakespeare’s play A Midsummer Night's Dream is a play about dreams. Dreams can be magical and playful, serious, and indulging. In some of the scenes in the play, the dreams are very sexual in nature as the fairies dance and cavort. In the play the dream that we call love is more natural and sensual like animal nature. The dreams help demonstrate desire. Bottom's dreams slip away as he returns to reality. The players move in between what is real and their dreams. Most of the dreams in the play symbolize love, desire, and dreams are used to predict the future. The outcome is not really affected by the dreams, but the dreams demonstrate the characters own interpretations of their emotions.