From fang, denoting a human tooth, and farrier, denoting someone who shoes or treats horses, the slang expression fang-farrier denotes a dentist.
—Cf., on the same pattern, the jocular expression spud-barber, denoting one who peels potatoes.

There have been earlier synonymous slang expressions:

1-: fang-faker, recorded in Slang and its Analogues Past and Present ([London]: Printed for subscribers only, 1891), by the British lexicographer John Stephen Farmer (1854-1916) and the British author William Ernest Henley (1849-1903):

Fang-faker, subs. (common).—A dentist. [From Fang, a long pointed tooth + Faker (q.v.) ¹.]

¹ These are the relevant definitions of fake and faker, from Slang and its Analogues Past and Present:

Fake, verb (common).—I. To do anything; to fabricate; to cheat; to deceive, or devise falsely; to steal; to forge. A general verb-of-all-work.
Faker, subs. (common).—I. One who makes, does, or ‘fakes’ anything.

2-: fang-wrencher, used (perhaps coined) by the English author Pelham Grenville Wodehouse (1881-1975) in Laughing Gas (London: Herbert Jenkins Ltd, 1936)—the following is about two dentists, I. J. Zizzbaum and B. K. Burwash:

Possibly because they were old dental college chums, or possibly from motives of economy, these two fang-wrenchers shared a common waiting-room.

The expression fang-farrier originated in the slang of the British armed forces during the Second World War. John Leslie Hunt and Alan George Pringle recorded it in Service Slang: A First Selection (London: Faber and Faber Limited, 1943), a dictionary of slang used in the British Army, Royal Navy and Royal Air Force:

Fang farrier, dentist.

The expression also occurs in Warrior Slang, published in The Halifax Daily Courier and Guardian (Halifax, Yorkshire, England) of Monday 23^rd August 1943:

If the fighting services in this war are not quite so strong on war songs as were those in the last it may be largely due to the fact of mechanisation and the consequent lack of the same route-marching incentive to rhythmic melody. But apparently they make up for this by developing far more extensive service slang. After this war, civilians may be confounded by the conversation of the ex-service youth of both sexes, for the women auxiliaries are picking up most of their masculine comrades’ slang.
Even those of the over-age brigade know what a “canteen medal” is. It is old Regular Army slang for a beer stain on one’s tunic. But what is a “Joe Soap”? It indicates a “dumb” or unintelligent person. A “Midwaaf,” on the other hand, is just an over-officious A.T.S. ² non-com. A “Zizz” is a rest period or a slacker’s paradise. One excellent invention is “Attery,” which is the living quarters, of course, of the A.T.S. Sausages, which were “bangers” to all old-school-tie disciples, become in the service slang “barkers,” with pretty obvious innuendo, and machine-gun bullets are “confetti,” the latter a pretty descriptive touch. An autogyro is an “egg-whisk,” and a dentist is a “fang farrier.”

² A.T.S.: the Auxiliary Territorial Service, i.e., the women’s branch of the British Army during the Second World War.

The expression fang-farrier was soon adopted into Australian English. For example, the following are two extracts from the column Peepshow, by Kirwan Ward, published in the Daily News (Perth, Western Australia):

1-: Of Friday 20^th August 1948:

You can’t keep a Clueless Chloe out of the papers for long. As soon as she feels that she’s being neglected by the typewriter tribe she turns on some feat of dizziness to attract attention.
She bobbed up again in a dentist’s surgery on Wednesday (August 18 in case you care), when she was standing her watch in the surgery of a local fang farrier.
In came a character wearing that apprehensive air common to dental patients and English batsmen on their way to keep an appointment with Lindwall ³.
“Is my appointment for 6 p.m. today?”
Chloe whizzed through the book. “No,” she said. “It’s for 6 p.m. Wednesday.”
“Okay,” muttered the character dazedly, and took his departure.

³ Ray Lindwall (1921-1996), who played in Australia’s national cricket team from 1946 to 1960, was one of the greatest fast bowlers of all time.

2-: Of Saturday 16^th October 1948:

If you’ve ever had any dental trouble you don’t have to be told that somewhere in the base of every human tooth there’s a molecule-sized imp whose one ambition is to find an exposed nerve and start pounding it with a sledge hammer.
Well, one of these imps went to work on some poor sailorman from Naval Staff office, Fremantle, and gave him the full treatment.
The tortured tar put on a stoical Nelson act (or anyway a half-nelson) but eventually had to totter along to the fang farrier who excavated the tooth and evicted the imp, with calm efficiency.
Bearing the tooth carefully back to Staff office the sailor propped it on his desk, sprinkled it with sugar and said “Now you . . . (here he used a coarse sailorly oath) ache as much as you like.”

In fact, the expression fang-carrier came to be regarded as Australian English—as illustrated by Lady Alice can give the dinkum oil, published in The Yorkshire Evening Post (Leeds, Yorkshire, England) of Friday 20^th November 1953:

When the Queen and the Duke of Edinburgh reach Australia they will have one qualified interpreter in the party. Lady Alice Egerton can give them the dinkum oil on the local lingo.
Lady Alice, 30-year-old sister of the Earl of Ellesmere, and one of the Queen’s Ladies-in-Waiting, has been studying a book of Australian slang sent to her by a friend.
She has already been trying out phrases like “amen-snorter” for a chaplain, and “chalk and talker” for a teacher. On an Australian sheep station she will know that a “jumbuck” is a sheep, and that the “baitlayer” is the station cook.
The task Lady Alice has set herself is a rewarding one. For the Australian idiom is probably the most vigorous and colourful in the English language.
In Australia you are not just short of money. You are “stone motherless broke” and “stiff as a crutch.”
Taking a rest is spine-bashing. A dentist is a fang-farrier. And a party with plenty to drink is a shivaroo with bags of stagger-juice.
That nondescript unpleasant character at the next table is a ratbag, and the spineless creature next door is “Three-penn’orth of God Help Us.”
When something goes wrong, to the Aussies it’s “a fair cow.” And if the thing is a complete failure, you score “a blue duck.”
There will he hundreds of receptions, hours of hand-shaking, weeks of ceremonial on the Australian tour. But a few well-timed “Good-ohs” and “Bonzers” from Lady Alice will do more for the bonds of Empire than all the Mayoral speeches lumped together.
As the Australians would say: “Good on you, sport. You’re pure merino.”
FOOTNOTE.—Dinkum oil—Aussie for accurate or genuine information. Lingo—language.