Reality itself leads to the downfall of Gatsby's dream. What he dreamed of was ultimately unattainable.
Gatsby's dream was not merely to reunite with Daisy. He wanted to erase the last five years and go back to the moment in 1917 when they were in love with each other. He wanted to start over as if nothing had happened in the meantime.
However, as Nick tries to explain to him, that's not possible. There is no going back. Too much has happened in between. Daisy has married and had a child. She's built a life separate from Gatsby. Gatsby may refuse to hear this, but it is nevertheless true.
In the end, one person can't be forced to fulfill another's fantasies. Daisy can't become what Gatsby wants. As we watch him struggle with his dream slipping away (something he never accepts), we come to realize that it was the dream itself, more than Daisy, that kept Gatsby going.
On a more practical plot level, Tom interferes with the dream. He too has a stake in how events unfold. He too wants Daisy as his wife. He has a hold on her, for they have been married for five years and have knit together as a couple, even if unhappily. Daisy doesn't necessarily want to break up the life they've built. Tom is also aggressive and brutal, and he undermines Gatsby in front of Daisy as a con man, a criminal, and a non-Nordic. He knows how to play on Daisy's emotions and how to win this battle.
The dream was doomed before the car accident: that incident simply hastened the end.
Jay Gatsby's dream was the quintessential American dream - success (money and riches) through whatever means one deems necessary. In this case, Gatsby began a penniless but determined to make his life better than that of his parents:
"His parents were shiftless and unsuccessful farm people - his imagination had never really accepted them as his parents at all. The truth was that Jay Gatsby, of West Egg, Long Island, sprang from his Platonic conception of himself" (104).
To him, his parents were unsuccessful and surely they could not really be his parents when he was destined or greater things. Through bootlegging, Gatsby made his riches. He became well-known and people came to his house every weekend for huge elaborate parties.
But, Gatsby's downfall was this exact same thing - success through whatever means one deems necessary. He did not honestly earn the money, and he spent his life trying to cover up lies and make himself seem great. Daisy was the one part of his American dream that he did not yet have, but fter Tom and Gatsby have their great argument in the city, Gatsby is sure he has just snagged her. However, Gatsby's penchant to lie about everything is his ultimate downfall. He lies about who was driving the car, which means that Tom is willing to tell Wilson that the car belongs to Gatsby. And just like that, Daisy escapes without punishment - and she still has a man - but Gatsby has lost everything.
After his death, we see the downfall and destruction of his American dream. His unsavory work-friends refuse to get 'mixed-up' in his death. The people who came to his parties were people that he did not actually know and they did not come to the funeral. Tom and Daisy left town after it happened, so the girl Gatsby wanted did not even come to the funeral. The only person who did come was Gatsby's father, a man whom Gatsby despaired of because he was poor and nonrepresentational of the American Dream.
Dishonesty is the downfall of Gatsby's American Dream, as it is the destruction of everyone else's - except Nick's - dreams as well. After all, as Nick stated in chapter 3:
"I am one of the few honest people that I have ever known" (p. 64)