The answer to your question can be found in the chapter called "How Am I Supposed to Write Poetry?." The first answer we see to this question is the one that Andy tells his English teacher, which is a lie. According to Andy's version of the story, he isn't handing in his poem because he hasn't written one. Having read Andy's poem at the beginning of the chapter, however, we as readers already know that this is not true.

The teacher gives him a hard time, telling him that "this was the last big grade for this quarter" and reprimanding him over the fact that he has failed to turn in numerous homework assignments in recent weeks. She acknowledges that he hasn't been feeling well in light of everything that he is going through, but also points out that this was "an easy assignment."

We discover the real reason why Andy didn't turn in his poem when he is talking to Keisha after class. When she tells him that she thought he had done the assignment, he admits that he did, but states that he just doesn't care, and didn't feel like turning it in.

This lack of caring could arguably have been seen as a warning sign of Andy's depressed state of mind, and if his teacher had read the poem, she surely would have heeded the warning contained in the words that Andy wrote.

I cannot see the future
And I cannot change the past
But the present is so heavy
I don’t think I’m going to last.

Tragically, the poem that Andy never turned in foreshadows his suicide. It is a clear indication that Andy is not coping in the aftermath of the accident that killed his friend, Robbie Washington.