The following slang expressions have been used to designate the mouth:
1-: box of ivories; also ivory-box;
2-: box of dominoes; also domino-box;
These are the earliest occurrences of those expressions that I have found:
1.1-: box of ivories: From Boxing. Great Fight between Hickman (the Gas-Light Man) and Neate, from Bristol, for 200 Guineas a-side, published in the Oxford University and City Herald (Oxford, Oxfordshire, England) of Saturday 15th December 1821:
Hickman came laughing to the scratch, full of confidence; but on his endeavouring to plant his tremendous right-handed hit on the throat of his antagonist, the length of Neate prevented it, and the blow alighted again on his shoulder: the Gas endeavoured again to make it, when the Bristol Hero gave Hickman so hard a blow on his box of ivories, that he retreated, and was also compelled to make a pause before he again commenced the attack.
1.2.1-: Perhaps in the following from The Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Tuesday 25th August 1730:
The Mystery of FREE MASONRY.
Question. Are you a Mason? Answer. I am.
Q. Is there a Key for our Lodge? A. Yes, there is.
Q. Where is it kept? A. In an Ivory Box between my Tongue and my Teeth, or under the Lap of my Liver, where the Secrets of my Heart are.
1.2.2-: ivory-box: From Greenwich Fair, a letter to the Editor published in The Morning Herald (London, England) of Wednesday 29th May 1822:
There was a glorious set-to, near Deptford, between two lily-whites. A frail fair one was the occasion of the scratch. It was a woman that caused the ten years’ siege of Troy—and it was a woman that raised a ten-round milling-match between the chieftains of colour—
“A bold virago, strong, and tall,
As Joan of France, or English Moll.”
She supported her champion in her capacious lap, wiped the dust from his sable brow, and kept up his pluck. Receiving, however, some damage in his ivory-box, and a few dowsers in his glimmers, he gave up the battle.
2.1-: box of dominoes: From Devon Wrestling in London, published in Woolmer’s Exeter and Plymouth Gazette (Exeter, Devon, England) of Saturday 16th June 1827:
Abraham looked steadily on his opponent, who was taller than himself, having a pair of stout “toddlers,” a good “victualling office” and “bread basket,” a regular box of “dominoes,” full “gills,” and piercing “ogles.”
2.2-: domino-box: From Boxing, published in Bell’s Life in London, and Sporting Chronicle (London, England) of Sunday 7th September 1823:
Fairchild came laughing up to the scratch, giving the wink to friends it was all right, when he hit Todd on the domino box. Both down.
3-: bone-box: From A Classical Dictionary of the Vulgar Tongue (London: S. Hooper, 1785), by the English antiquary and lexicographer Francis Grose (1731-1791):
Bone Box, the mouth. Shut your bone box; shut your mouth.
4-: potato-box: From Fight between Ned Neal and Roche, published in The Star (London, England) of Wednesday 3rd December 1828:
Ned popped in a heavy body hit with his left, and broke away—again to business, he made his right tell on poor Roche’s potato box, and then letting fly his right with immense force dropped him on his head.
5-: potato-jaw: From the diary of the English author Frances d’Albray (née Burney – 1752-1840)—as published in Diary and Letters of Madame d’Arblay (London: Published for Henry Colburn, by his successors, Hurst, and Blackett, 1854):
[4th June 1791] Mrs. Schwellenberg, who had sat laughing and happy all this time, now grew alarmed, and said, “Your Royal Highness, I am afraid for the ball!”
“Hold you your potato-jaw, my dear,” cried the Duke, patting her.
6-: potato-trap: From A Classical Dictionary of the Vulgar Tongue (London: S. Hooper, 1785), by the English antiquary and lexicographer Francis Grose (1731-1791):
Flash, […] to shew ostentatiously; to flash one’s ivory, to laugh and shew one’s teeth; don’t flash your ivory but shut your potatoe trap and keep your guts warm, the devil loves hot tripes.
Red rag, the tongue; shut your potatoe trap, and give your red rag a holiday, i.e. shut your mouth, and let your tongue rest; too much of the red rag, too much tongue.
7-: kissing-trap: From The Adventures of Mr. Verdant Green, an Oxford Freshman (London: Nathaniel Cooke, 1853), by Cuthbert Bede, pen name of the English novelist and Church of England clergyman Edward Bradley (1827-1889):
After many refusals, our hero was at length persuaded to put on the gloves, and have a friendly bout with Mr. Blades. The result was as might have been anticipated; and Mr. Smalls doubtless gave a very correct résumé of the proceeding (for, as we have before said, he was thoroughly conversant with the sporting slang of Tintinnabulum’s Life), when he told Verdant, that his claret had been repeatedly tapped, his bread-basket walked into, his day-lights darkened, his ivories rattled, his nozzle barked, his whisker-bed napped heavily, his kissing-trap countered, his ribs roasted, his nut spanked, and his whole person put in chancery, stung, bruised, fibbed, propped, fiddled, slogged, and otherwise ill-treated. So it is hardly to be wondered at if Mr. Verdant Green from thenceforth gave up boxing, as a senseless and ungentlemanly amusement.
Illustration by Cuthbert Bede, from The Adventures of Mr. Verdant Green, an Oxford Freshman (London: Nathaniel Cooke, 1853):