I would say no. I do not feel like his use of hyperbole makes the poem seem insincere. I think the main reason for this is that hyperbole is really quite common in poetry and so it just seems normal. If someone were to say this stuff out loud, it would seem weird and insincere, but in poetry it does not really.
The other thing is that most of his hyperbole is about how long he'll love her. And you can argue that saying you'll love someone eternally is not hyperbolic if you believe in eternal life after death.
Towards the end of his life Robert Burns (1759-96) was engaged in the task of collecting old traditional scottish songs. In one of his letters in 1794 he states that the song "A Red, Red Rose" is "a simple old Scots song which I picked up in the country." Hence the poem is not his original composition.
Burns' poem, however, is charmingly simple and direct in its method of praising his lover and most significantly describes how much he loves her:"As fair art, thou my bonny lass/So deep in luve I am."
He then tells her how much he loves her:
I will luve thee still, my Dear,
While the sands o'life shall run.
The implication is that he will love her forever, that is, infinity. As long as human life exists on this earth he will love her. Burns uses hyperbole, that is, exaggeration to convey to his lover the depth and intensity of his love for her. In the previous line he has told her that he will love her till all the seas dry up! But he is not satisfied with that, because he feels that there is a possibility that all the seas may indeed dry up so he says that he will love her till all human life comes to an end on planet earth!
The use of hyperbole by Robert Burns is certainly not insincere because the use of such hyperbolical metaphors was an accepted convention in love poetry.
Everyone is familiar with this most commonplace of similes. But although the poem begins with two overt similes (my love is like a rose; my love is like a melody), it moves beyond them in the poem's three other quatrains. The second quatrain goes from a genuine simile to a comparison involving an "as" ("I am as much in love with you as you are fair"), which is sort of a simile, but sort of not. And the quatrain then moves into a rhetorical hyperbole ("I'll love you until all the seas run dry," etc.) which continues for six lines. The poem ends with another figure, this time one that combines hyperbole (exaggeration) and an implied simile ("my love is so strong that it can encompass vast space and time"). One thing to look for in any simile or metaphor is the essential proposition that things can resemble one another only if they are not identical. That is, x is like y because and only when x is not y. Difference is as important as similarity.
Thus, by using hyperbole, Burns ignores the commonplace implications of the simile of female beauty to a flower and makes it more emphatic and dramatic (i.e., that roses fade and so young maidens should make the most of their youthful energy and give themselves to their lovers immediately).
Hyperbole is "obvious extravagant exaggeration or overstatement... used figuratively to create humor or emphasis" (Kathleen Morner & Ralph Rausch, NTC's Dictionary of Literary Terms). Hyperbole is mostly used in love poems. This literary device is a wonderful way do express inner feelings through written or printed words.
Not necessarily hyperbole makes comparison insincere or fake. Rather, it is a nice figurative device to bring a dramatic effect, to emphasize on the certain topic and make the language ornamented. Same proclamations are applicable for the above poem. Moreover, the poem has a background which is related to Robert Burns' personal life. And this factual background would surely make you convinced that the hyperbole is only exaggeration if read as written cluster of letters, but is actually much more than that if thought with heart keeping the background in mind.
The famous Scottish poet Robert Burns (1759-1796) felt in love with Jean Armour who was from a conservative family, and their love affair was a very deep one. When Ms. Armour was found pregnant, her father outrageously discarded Robert, and disheartened Robert sailed to West Indies. To bear the expense of the voyage, he decided to publish his poems.
During his stay in Edinburgh, Robert Burns met printer James Johnson, who planned a project to print all of the folk songs in Scotland. This project enthralled Burns and embarked upon a journey throughout Scotland to collect as many folk songs as possible. Burns collected over 300 songs and wrote a few himself, including "A Red, Red Rose."
The poem 'A Red, Red Rose' was a tribute to his own love. He wonderfully depicted his deep, loyal love towards his beloved through the poem. His use of hyperbolic similes are the rich and true expressions of his love, the love for which he endured huge torments, but finally, Jean's father accepted him, and he married his beloved.