In the book The Outsiders, what is the significance of Robert Frost's poem?

Robert Frost's poem "Nothing Gold Can Stay" plays two significant roles in S.E. Hinton's novel The Outsiders.
First of all, the poem can be analyzed as a meditation on youth and innocence, both states of existence that are impermanent. Youth and innocence are also thematically important to the novel as all of the greaser boys described in the novel are old before their time. Because the conditions of their lives force them to grow up sooner than perhaps they would choose otherwise, the boys are unable to hold on to their innocence. Adult worries plague all of them, though they are all very young, and traumas abound, ranging from the deaths of beloved parents to emotional abuse to physical beatings.
As well, the poem is important to the plot of the novel. Ponyboy recites "Nothing Gold Can Stay" to Johnny while they are hiding out in the church. The beauty and depth of the poem makes immediate sense to Johnny, and he later implores Ponyboy to "stay gold" from his deathbed. This moment marks one of the most poignant in the entire novel, as Johnny succumbs to his injuries and begs Ponyboy to maintain his innocence and goodness in the face of violence and poverty. Johnny's attachment to the poem elevates it from a school assignment to an anthem for the underdog.

The Robert Frost poem that the question is alluding to is "Nothing Gold Can Stay." In a nutshell, the poem is saying that nature's seasonal beauty will always come to an end. Nature's "gold" is wonderful and beautiful; however, it simply can't always stay that way forever.
When Johnny is dying in the hospital, he tells Ponyboy to "stay gold." Johnny recognizes that Ponyboy is different than the other Greasers. The other Greasers know this too, and that's why they are constantly trying to protect him from the harshest realities of their gang life. Johnny, Soda, and Darry all recognize that Ponyboy has huge potential to leave this current life behind. He's a dreamer, and he's a smart dreamer too. This is also why Cherry is so taken with Ponyboy as well. She knows that he is different and special. He is "gold." Johnny doesn't want the Greaser life to taint that part of Ponyboy. This is why, in his dying breath, Johnny tells Ponyboy to stay gold. Johnny believes that Ponyboy can do what Frost says is impossible.