I wanted to point out an earlier instance of smithereens than the Phrase Finders' claimed 1801 occurrence in Francis Plowden, The History of Ireland, cited in user66974's answer.
From William Macready, The Bank Note, or, Lessons for Ladies: A Comedy in Five Acts (London, 1795):
Servant. A letter from Mrs. Flounce.
Sally. Let's see.—(exit Servant.)—Oh! what a seal!—a heart stuck with darts, like pins in a pincushion. (Reads) "Thou fairest of the fair.——I send you this by the Penny-Post,—and if the letter carrier does not give it to you directly, run to the Post-office to enquire for it, and then we'll be sure to meet, for i am going to your house immediately, where I hope you'll receive with exstacy, your ever agreeable and transported, William Killeavy." Pshaw!—what signifies transported, it would be something indeed if he had been hang'd,—or shot,—or——
Enter Killeavy.
Killeavy. I wish he was with all my heart.—If the rascal is troublesome to you,—who is he, my pet?
Sally. Who? Why, a person who pretends to love me.
Killeavy. Oh, then I join you with all my heart,—and wish he was hang'd, shot, cut in smithereens, or—
Sally. Ha! ha! ha! that’s a very good Joke,—why it was yourself I meant.
In the play, Killeavy is an Irish servant to a character named Sir Charles Leslie.
Jonathon Green, Chambers Slang Dictionary (2008) confirms that the form smithereens originated in Ireland:
smithereens n. (also flindereens) {SMITHERS n. + Irish dimin. -een, Share suggests Irish smiodar, a fragment} {mid-19C+} tiny fragments, atoms; esp. in phr. smashed to smithereens, blow/break/knock/split to/into smithereens, smashed to pieces, often in fig. use.
I had no luck finding eighteenth-century instances of smithers or flindereens, however.
Although "smashing to smithereens" sounds like an inherently violent process, Green is correct that the expression may be used figuratively to refer to such things as emotional tumults that involve no real-world violence. One such early example is from "The Can-Can," in The Book of Comic Songs and Recitations (1874):
A girl who lives up our Court, / Served me as she didn't ought; / And made of me a cruel sport, / Though after her I ran. /She knocked my heart to smithereens, / All for a chap of larger means, / Who made Java Coffee out of beans, / While she danced the Can-Can.
With regard to the Phrase Finder date of 1801 for smithereens in Francis Plowden, The History of Ireland, the quoted text actually appears in Plowden's The History of Ireland, From Its Union with Great Britain, in January 1801, to October 1810, volume 3 (1811), which reports that the word was used in a threatening anonymous note left near the hall door of a magistrate named Pounden in the summer of 1810:
The form of this notice was. "Mr. Pounden, Sir, we gave you notice some ago to quit this country ; for you are making a rebellion here. We tell you now again, that if you don't be off directly, by the ghost of William, our deliverer, and by the orange we wear, we will break your carriage in smithereens and hough your cattle, and burn your house—so mind yourself—you will soon hear again from your friend, TRUE BLUE"
It thus appears that the notice cited in The History of Ireland was written by night marauders in the summer of 1810 and reprinted in Plowden's history in 1811. The 1795 instance from Macready's play is 15 years older.