The phrase to make the fur fly means: to cause trouble or an argument.

This phrase occurs, for example, in an article by Jessica Duchen about the U.S. theatre director Peter Sellars (born 1957), who was then the English National Opera’s director-in-residence, published in The Independent (London, England) of Wednesday 25^th February 2015:

The Indian Queen’s score was left unfinished on Purcell’s death, aged 36; its original play by John Dryden is, according to Sellars, “unrevivable”. He has replaced it with a spoken text drawing on the Nicaraguan novelist Rosario Aguilar’s The Lost Chronicles of Terra Firma, telling of the Spanish invasion of the Mayan people from the women’s point of view; and to bridge musical gaps he has incorporated music from Purcell’s religious anthems and theatrical songs.
This reconstructive approach could make the fur fly among purists in the composer’s native UK, but Sellars is not worried.

The texts containing the earliest occurrences of the phrase to make the fur fly that I have found:
– indicate that this phrase is based on the image of cats fighting—cf. to fight like Kilkenny cats;
– relate to the war which opposed the USA and the United Kingdom from June 1812 to December 1814.

These early occurrences are as follows, in chronological order:

1-: From a letter, dated Chateauguay, Monday 4^th October 1813, that Major-General Wade Hampton (early 1750s-1835) wrote to the U.S. Secretary of War, John Armstrong Jr. (1758-1843)—letter published in The National Advocate (New York City, New York, USA) of Tuesday 22^nd February 1814:

There has been inculcated by the artifices of the British, a shameful and corrupt neutrality on the lines, for the purposes of gain. I have directed these officers to break the truce. And should other means fail, to act the part of the mischievous urchin, who, to get two peaceable tabbies at “making the fur fly,” hold [sic] them up together by the tail.

2-: From the Plattsburgh Republican (Plattsburgh, New York, USA) of Saturday 12^th March 1814:

It is with pleasure we inform our readers, that General Wilkinson seems determined to destroy the traitorous intercourse kept up, by men who call themselves Americans, with our enemies in Canada: Small detachments have been tried without effect, and now strong ones are put in motion.—Colonel Clark, Old Rifle, marched the 8^th instant, with Maj. Bayley, and a detachment of one thousand infantry and one hundred mounted rifle-men, all Green Mountain Boys, to take possession of the frontier, from the lake east to Connecticut river; and on the 10^th inst. another detachment of 300 prime riflemen and sixty dragoons, marched under Major Forsythe, whose name carries terror to the enemy, to guard the lines west of the Lake.
We understand the orders of those officers are to make prisoners every British subject detected within the limits of the United States, and to apprehend and deliver to the civil authority, for trial and punishment, every American citizen found in Canada—therefore smugglers look out, or you will soon see “the fur fly.”

3-: From the Salem Gazette (Salem, Massachusetts, USA) of Tuesday 15^th March 1814:

The American and Canadian citizens who reside near the lines, and who are closely connected by kindred and personal friendship, had mutually agreed to abstain from hostilities and depredations. This mode of proceeding however our Generals would not tolerate, and they endeavored to drive them to bloodshed by the following dignified expedient, as is related in a Military Letter from Gen. Hampton to the Secretary of War. Hampton says,
“I have directed my officers to break the truce; and, should other means fail, to act the part of the mischievous urchin, who, to get two peaceable tabbies at “making the fur fly,” hold [sic] them up together by the tail.”
What an improvement in the art of War! Our Generals, it must be confessed, much resemble these mousers, for they are “peaceable tabbies,”—have made a horrible and heart-rending caterwauling in their Proclamations,—and are now “making the fur fly” by being themselves “held up together by the tail.”

4-: From the Salem Gazette (Salem, Massachusetts, USA) of Tuesday 29^th March 1814:

“The Newburyport folks cannot be content that our brave seamen should be paid for their services. They are called freebooters, and any name but honest men. It is suspected that some talk nearer home has as much envy as it has political prejudice in it.” Register.
Mr. Bentley probably means, by “some talk nearer home,” to allude to the hard names and severe epithets which Mr. DEXTER last summer applied in the District Court to those privateersmen who sailed from this port for the sole purpose of capturing the property of our own citizens. No man could use harder language: he called them all to nought. Gov. Dexter, in the course of his “candid discussion,” declared in open Court, they were as bad as “Algerine corsairs or Sallee rovers,” and he said that Salem was in danger of becoming “New Providence or Dunkirk.”
Now if the Silver Grey ‘folks’ had said any thing half as severe, we should admire to see brother Bentley attack them pell-mell, and make the fur fly: but as for Mr. Dexter, he is not to be regarded, as Hone would say, because “he is always defending smugglers.”

I have found an early occurrence of fur-flying (referring to the above-quoted letter by Wade Hampton) in the remarks that a person signing themself ‘Vidette’ made on a declaration by the U.S. Secretary of War, John Armstrong Jr.—published in the Political and Commercial Register (Philadelphia, Pennsylvania, USA) of Wednesday 8^th June 1814:

“Washington carried his whole force against the Hessians in New Jersey, and beating them, recovered that moral strength, that self confidence, which he had lost by many preceding disasters. We are now in that state of prostration which he was in, after he crossed the Delaware, but like him—We may soon—get on our legs again!”—Armstrong.
Like him!—We!!—Like Washington!!!—O good sirs!—Nay Mr. Secretary—Not so fast, gentlemen—There is no similitude—The comparison is odious […].
[…]
“Like Washington!”—He was no tail wounding, tree girdling, back-door closing, step-back treading, out-rooting, Upper Canada-line sweeping, eyes right eyes left en potence appuying, stopper pulling, pole-straddling, caterwauling, fur-flying, mischievous urchin.