M-W provided this background on the word:

In the 5th century B.C. there was a rabble-rousing Athenian, a politician named Hyperbolus, who often made exaggerated promises and claims that whipped people into a frenzy. But even though it sounds appropriate, Hyperbolus' name did not play a role in the development of the modern English word "hyperbole." That noun does come to us from Greek (by way of Latin), but from the Greek verb "hyperballein," meaning "to exceed," not from the name of the Athenian demagogue.

A few passages come to mind:

But let your statement be, 'Yes, yes' or 'No, no'; anything beyond these is of evil. (Matthew 5:37)

Messiah tells us to be direct in our statements.  I take this as a caution against hyperbole.

Another passage that comes to mind is in Matthew 5:

For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished.  Whoever then annuls one of the least of these commandments, and teaches others to do the same, shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven.  (Matthew 5:18-19)

Was Messiah speaking hyperbole or was He speaking honestly and directly?

Here is the The American Heritage Dictionary definition of the English word annul:

1. 1. To make or declare void or invalid, as a marriage or a law; nullify.
   2. To obliterate the effect or existence of: "The significance of the past . . . is annulled in idle gusts of electronic massacre" (Alexander Cockburn).

Let us consider the word "annul" used in this passage.  It is the Greek word luon (Strong's #3089).  The Strong's Greek Dictionary defines the word this way:

A primary verb; to “loosen” (literally or figuratively): - break (up), destroy, dissolve, (un-) loose, melt, put off.

Let's examine the passage and consider His words:

For truly I say to you     [sounds like this is serious stuff... not hyperbole] until heaven and earth pass away     [last time I checked they are still here :)] not the smallest letter or stroke shall pass from the Law until all is accomplished     [apparently all is not accomplished... heaven and earth are still here] Whoever then annuls     [nullifies, declares invalid, loosens, destroys] one of the least of these commandments     [traditionally Deuteronomy 22:6-7] and teaches others to do the same     [the second part of the two-part condition] shall be called least in the kingdom of heaven     ["least" here is the Greek word elaxistos which does indeed mean "least"] but whoever keeps and teaches them     [again a two-part condition] he shall be called great in the kingdom of heaven.

Note that "these commandments" in this passage never refers to the Ten Commandments but to "the Law" with its (traditionally) 613 commandments.

Serious food for thought... not hyperbole.

Now to the King eternal, immortal, invisible, the only God, be honor and glory forever and ever. Amen. - 1 Tim 1:17