What function or purpose does the poetry in the novel Hongloumeng (Dream of the Red Chamber) serve in the story's overarching narrative? Is it simply a plot device?

Poetry in Dream of the Red Chamber serves as a major plot device, but it also serves as an extension of each character. The three most important characters, Jia Baoyu, Xue Baochai, and Lin Daiyu, are the three most talented poets. As the novel progresses, it becomes clear that the often ill and emotional Lin Daiyu is the most accomplished. This sets the novel up for the interesting dynamics between the three characters' importance to one another.
Jia Baoyu (the "Precious Jade") and Lin Daiyu (the reincarnated "Crimson Pearl Flower") both come from the fairy world. The only two otherworldly beings in the novel out of a cast of nearly four hundred characters, they are able to recognize this in each other through poetry. Jia Baoyu is always incredibly taken with Lin Dauyu's poems, which are often unbearably sad and which reflect her extreme moodiness and hint at her mystical origins and tragic fate.
The poems of Xue Baochai, in contrast, show her grace and composure. Xue Baochai is textbook perfect. While her poems are not as good as Lin Daiyu's and Jia Baoyu is not drawn to her in the way he is to Lin Daiyu, it is Xue Baochai who becomes his wife.
While books that include poems usually include them to describe scenery or perhaps express sentiments for another character, Dream of the Red Chamber does this a little differently. It is extremely poem-heavy, but each poem is important to the story as an expression of each character's inner workings. For example, the characters meet often for structured poetry contests and write on the same subject. The poems they write on chrysanthemums, snow, or the moon reveal the characters' thought processes and personalities.