What is the role and idea of nature in the story "Rappaccini's Daughter" by Hawthrone?

Excellent question.  "Rappaccini's Daughter" is actually a commentary both on the nature of man and the nature of science.  The science, here, comes in the form of manipulating humans and nature for one's own purposes.  Let's stick with nature side of things to answer your question.

In this garden, which contains all the nature we know of in this story, there is both life and death. The flowers are beautiful and lush and colorful; however, the owner of the garden is not able to enjoy their beauty...

the man's demeanor was that of one walking among malignant influences, such as savage beasts, or deadly snakes, or evil spirits, which, should he allow them one moment of license, would wreak upon him some terrible fatality.

We know, of course, that this garden is full of poisonous plants.  They seem look beautiful, but to touch them or smell them could be death.  Dr. Rappaccini has taken something lovely and turned it into something deadly--all for the sake of experiment.  Nature, then, has taken on the role of a character in this story, one which can inflict harm and even kill.

If the doctor had just confined his work to the plants and flowers, no real harm would have been done.  Instead, we soon discover he has infused not only the flowers with the poison but humans, as well--including his own daughter.  Beatrice seems to be at least content with things as they are; however, Giovanni, once he discovers what has been done to him, is not so accepting. He see the monstrousness of this once-beautiful garden.

The aspect of one and all of them dissatisfied him; their gorgeousness seemed fierce, passionate, and even unnatural. There was hardly an individual shrub which a wanderer, straying by himself through a forest, would not have been startled to find growing wild, as if an unearthly face had glared at him out of the thicket.

He says, in fact, that many of them don't even look like they could have been part of God's creation.  And that's the point.  Neither plants nor people were intended to be manipulated in such a way as to lose their essential identity.  Creating poison where there should have been life and beauty is evil.  Doing the same for humans is beyond evil.