What does the green light and Doctor Eckleburg's eyes symbolize?
In The Great Gatsby, green symbolizes: the American dream, money, go, envy, Daisy, youth, spring, fertility, inexperience, boyhood. Gatsby worships the green light: Nick observes him bowing down to it. It is the confluence of all of Gatsby's hopes and dreams and his desire to retreat to the past, to when America was the young, fertile, all-possible land.
Gatsby equates the color to Daisy, as she embodies Gatsby's desire to repeat the past. He covets his first love, but he does not realize that she is corrupt (maybe she was always corrupt). So his pursuit of all that is green is a false quest, and it will inevitably end in failure. Gatsby fails to realize that Daisy doesn't symbolize any of these things: Daisy's voice is full of money, but she doesn't say anything; she's not fertile (she's a terrible mother); he cannot recapture his youth (the past cannot be repeated). Notice the contrast of "fresh green breast of the new world" to Myrtle's flapping, destroyed breast after she is hit by Daisy. The pursuit of the green light results in slaughter.
The Eyes of Dr. T. J. Eckleburg symbolize the deus abscondi world: a world in which God is hidden or removed. In this kind of world, God cannot be perceived or seen; He works in revelation (symbols, omens, colors). God has been removed from Gatsby's world, and he leaves this billboard to serve as such. Notice that it is placed in the Valley of Ashes: God is revealing that the world is a corrupt place like Sodom and Gomorrah (also destroyed by fire: left in ashes). Fitzgerald's morality is indirect but clear: materialism leads to carelessness and death.
Of course, this is just opinion, because those things do not really have to symbolize anything. But in my opinion, the green light symbolizes all the things that Gatsby hopes for. In other words, it symbolizes Daisy since wants her more than anything else. After Gatsby is dead, in the last chapter, Nick thinks about the light in more general terms, but it is still about dreams. Now he is thinking about it in terms of the dream that is America.
The Doctor's eyes presumably symbolize the eyes of God, watching everything we do. Of course, in Chapter 8, Michaelis tells George Wilson that they are just a billboard. But they do seem to me (as to George) that the eyes are a symbol of God.