Three-dollar bills were printed by various states in the 1800s, but there has never been a U.S. three-dollar bill. There are U.S. one-dollar bills and U.S. two-dollar bills.
 
“Rufe Davis describes a certain ham: ‘Phoney as a Three Dollar Bill!!!’” was cited in Hy Gardner’s “Broadway Newsreel” column in the Brooklyn (NY) Daily Eagle on March 19, 1937. “Rare as a three-dollar bill” was cited in a 1942 newspaper and “queer as a three-dollar bill” in a 1943 newspaper. The expression “phon(e)y/queer/strange/rare as a three-dollar bill” means something unusual or fake.
 
“Queer/strange as a three-dollar bill” also referred to homosexuals by at least the 1960s.
   
   
Wikipedia: Fake denominations of United State currency
$3
Although various US states printed $3 bills before the unification of the currency, no US$3 bills have ever been printed. However, various fake US$3 bills have been released over time, generally poking fun at politicians or celebrities such as Richard Nixon, Michael Jackson, George W. Bush, both Bill and Hillary Clinton, and Barack Obama in reference to the idiomatic expression “queer as a three-dollar bill” or “phony as a three-dollar bill”. In the 1960s, Mad printed a $3 bill that featured a portrait of Alfred E. Neuman and read: “This is not legal tender—nor will a tenderizer help it.” Mad writer Frank Jacobs said that the magazine ran afoul of the US Secret Service because the $3 bill was accepted by change machines at casinos.
 
Dictionary.com
queer as a three-dollar bill
adverb
Obviously homosexual •Usu considered offensive
 
The Free Dictionary
*phony as a three-dollar billand *queer as a three-dollar bill
phony; bogus
 
Brooklyn Newsstand
19 March 1937, Brooklyn (NY) Daily Eagle, “Broadway Newsreel” by Hy Gardner, pg. 14, col. 7:
Rufe Davis describes a certain ham: “Phoney as a Three Dollar Bill!!!”
 
3 June 1942, Charleston (SC) Evening Post, “Today’s Sport Parade” by Jack Guenther (U.P.), pg. 9, col. 3:
Today the player who has held his position with the same team for as long as three seasons is as rare as a three-dollar bill.
 
19 September 1942, New York (NY) Age, “Time Out” by Buster Miller, pg. 11, col. 1:
That Dan Burley-Wendell Smith feud is as phony as a three-dollar bill.
 
24 September 1943, Vernon (TX) Daily Record, “News of Class 44-C,” pg. 7, col. 4:
“Three-dollar-Bill” Ferris never tells a man to shut his big fat trap—he phrases it this way, “Hush, now!! At ease, I say!!”—queer as a three-dollar bill.
 
6 October 1943, New York (NY) Times, “American Returns, Defies China Lure” by Brooks Atkinson, pg. 15, col. 3:
Transportation in China being as scarce as a three-dollar bill, the pressure to ride in these transports is tremendous ...
 
14 January 1945, Arkansas Gazette (Little Rock, AR), “Weary-Go-Round” by Hardy “Spider” Rowland, pg. 11, col. 7:
However, if anyone wants to wager that at least one of the accounts wasn’t wrong as a three dollar bill, please get in touch with me.
 
2 October 1945, Washington (DC) Post, “Planes Dropped Phony Money to Japs as War Ended,” pg. 4, col. 7:
Stage money—as phony as a three dollar bill—was one of the psychological weapons used by the Allied forces in the closing days of the Japanese war.
 
12 December 1946, Chicago (IL) Daily News, “In the Wake of the News” by Arch Ward, sec. 4, pg. 59, col. 2:
Add Similes
As phony as a three dollar bill.
—Frank D. Reid
 
8 September 1952, Boston (MA) Daily Globe, “The Caine Mutiny” by Herman Wouk, pg. 12, col. 3:
“Hell, don’t take anything Tom says too serious. Tom’s queer as a three-dollar bill.”
 
5 April 1953, Boston (MA) Sunday Advertiser, “Tommy COllins Puts i in Acton” by John Gillooly, pg. 29, col. 7:
“Some of these people, especially visiting reporters and photographers who don’t know me, must think I’m queer as a three-dollar bill.”
 
29 September 1957, Indianapolis (IN) Star, “Arkansas 1st Issued ‘Confederate’ Money” by Jim Buck, sec. 8, pg. 4, col. 4:
But strangely enough, the $3 bill which almost every state has printed, has never been approved by the Federal Government. Thus we find the expression, “strange as a three dollar bill.”
 
26 September 1975,  Courier-Post (Camden, NJ), “Life can be gay in the new Army” by Dennis M. Culnan, pg. 33, col. 3:
BOB SMILED to himself, mentally laughing about basic training. Here he was queer as a three dollar bill taking a shower with 60 other men…It’s a homosexual’s dream come true.
   
OCLC WorldCat record
Queer as a three dollar bill.
Author: Ken Wood, (Illustrator); Lesbian/Gay Freedom Day Committee (San Francisco, Calif.),
Publisher: San Francisco : Lesbian/Gay Freedom Day Committee, [1981] ©1981.
Edition/Format:   Image : Picture : Artwork reproduction : English
 
OCLC WorldCat record
Bring him back dead
Author: Day Keene
Publisher: Blue Ash, Ohio : Prologue Books, ©1984.
Series: Prologue Crime.
Edition/Format:   eBook : Document : Fiction : English
Database: WorldCat
Summary:
He was as rare as a three-dollar bill ... an honest man in the town of French Bayou - that was crowding Phenix City out of corruption’s first place.
   
Google Books
The New Partridge Dictionary of Slang and Unconventional English: J-Z
By Tom Dalzell and Terry Victor
New York, NY: Routledge
2006
Pg. 1955:
three-dollar bill noun
a homosexual US
from the expression “as strange as a three-dollar bill”
The Guild Dictionary of Homosexual Terms, p. 44, 1965
Dale Gordon, The Dominion Sex Dictionary, p. 156, 1967
 
Urban Dictionary
queer as a three dollar bill
Older term that means extremely strange but is usually heard in reference to a gay person. There never was a three dollar bill…thus the rarity.
(...)
by thedzone October 07, 2009