“To work one’s tail off” (also “to work/bust one’s ass” or “to bust one’s butt”) is to work strenuously. “Wales (also an International News Service staff correspondent) and I have been working our tails off” was cited in print in 1918.
 
   
(Oxford English Dictionary)
tail, n.
The lower and hinder part of the human body; the fundament, posteriors, buttocks, backside. tail over top = top over tail at top n.1 25d. Now dial. and colloq. (chiefly U.S., esp. in fig. phrases, as to work one’s tail off , to work strenuously).
1935   J. T. Farrell Judgment Day iv. 86   This idea of sweating your tail off with work..is the undiluted crap.
1942   W. Faulkner Go down, Moses 229   This is the first time you’ve had your tail out of that kitchen since we got here except to chop a little wood.
1969 New Yorker 14 June 72/3   Go out there and work your tail off. Don’t wake up tomorrow morning regretting that you didn’t give a hundred per cent.
1976   Billings (Montana) Gaz. 1 July 4- e/1,  I worked my tail off to help win a pennant for the Dodgers.
   
30 August 1918, Logansport (IN) Pharos-Reporter, “Getting blown out of cars, having Spanish ‘flu’ and stepping on dead Boche are just a few of the trials of I. N. S. correspondent” by Newton C. Parke (International News Service Staff Correspondent), pg. 6, co. 3:
Wales (also an International News Service staff correspondent) and I have been working our tails off and I hope we are getting some results, though I am just beginning to fall into the geography and general strategy of the thing up here.
   
Google Books
14 September 1918, The Fourth Estate, “Correspondents’ Job No Sinecure,” pg. 12, col. 1:
“Wales (also an International News Service staff correspondent) and I have been working our tails off and I hope we are getting some results, though I am just beginning to fall into the geography and general strategy of the thing up here.
 
Google Books
The Belt
By Paul Field Sifton
New York, NY: Macaulay Co.
1927
Pg. 146:
VOICES
Yeah!...He’s good all right!...Works our tails off!
 
Google Books
Studs Lonigan: A trilogy containing Young Lonigan,
The young manhood of Studs Lonigan, Judgment day
By James Thomas Farrell
New York, NY: The Modern library
1935
Pg. 78:
“You bet, Studs, this idea of sweating your tail off with work and carefulness is the undiluted crap.”
 
Google Books
Ever Since Eve:
A Comedy in Three Acts
By Florence Ryerson and Colin Clements
New York, NY: Samuel French
1941
Pg. 40:
SPUD. You betrayed me — that’s what about it! I work my tail off getting an exclusive on a new little number, and when I get here what do I find? You underselling me ... trying to steal my customer.
 
Google Books
Death and Taxes
By David Dodge
New York, NY: Macmillan Co.
1941
Pg. 172:
“I didn’t stay up all night working my tail off on this case so you could tell me what your ideas are about Marian Wolff.”
 
12 December 1943, The Plain Dealer (Cleveland, OH), “‘enery” by Leon Ware, This Week magazine, pg. 7, col. 3:
‘I’ve been working my tail off for this outfit, and now when the fun comes I get left.”