This is what is believed to be an exhaustive list of Jeopardy! episodes that have finished with no winner, as all players either finished with $0 or missed Final Jeopardy.
September 11, 1984 (Episode #2)
This game is found in J! Archive.
Greg Hopkins, the first champion in Jeopardy! history, was leading with $9,500. Lynne Crawford was in second at $5,000. Paul Schaffer had $1,100.
Final Jeopardy! category: THE CALENDAR
Final Jeopardy! clue: Calendar date with which the 20th century began (What was January 1, 1901?)
All three players wrote, “What is January 1, 1900?”, inexplicably bet everything—though the author suspects that the champion may have felt aggrieved by Alex having called him a chicken regarding his small Final Jeopardy! bet from a runaway position in episode #1 and wanted to make a point—and finished with $0. Three new contestants played episode 3.
April 8, 1985 (Episode #151)
This game is found in J! Archive.
This match was between John Voorhuis (a songwriter from Watertown, MA), Bill McKeever (a Naval officer and flight instructor from Virginia Beach, VA), and Ezra Litwak (a writer originally from Ann Arbor, MI, whose 2-day total was $6,399). Going into Final Jeopardy!, Bill and John had $2,500, while Ezra had $600.
Final Jeopardy! category: THE CALENDAR
Final Jeopardy! clue: In 1984, the Cotton, Fiesta, Orange, Rose, & Sugar Bowls were all played on this date (What is January 2, 1984?)
All three players responded with “What is January 1, 1984?”, bet everything, and finished with $0. The next episode saw the debut of early 5-time champion Mike Day.
(Thank you to Stefan Goodreau for the initial information about this Final Jeopardy. Further information comes from the National Archives of Game Show History.)
July 9, 1991 (Episode #1597, Seniors Tournament Quarterfinal #2)
This game is found in J! Archive.
Pat Van Melle and Florence Cannizzo were tied going into Final Jeopardy! at $5,900; Dow Huskey was in a distant third place at $2,100.
Final Jeopardy! category: STATE CAPITALS
Final Jeopardy! clue: 1 of 2 state capitals that formerly served as capital of the U.S.
None of the three contestants could come up with Annapolis, MD, or Trenton, NJ. Defending against each other, Pat and Florence bet all $5,900. Apparently believing he needed as much money as possible for a potential wild card, Dow also bet everything. Due to the tournament format, five wild card spots for the semifinals were awarded.
March 2, 1998 (Episode #3116, Celebrity game)
This game is found in J! Archive.
This was a Celebrity Jeopardy game.
Scores going into Final Jeopardy:
Jane Curtin $8,500
Teri Garr $2,700
Naomi Judd $900
Final Jeopardy! category: TV SITCOMS
Final Jeopardy! clue: For your information, this series that premiered in 1988 is TV’s longest-running sitcom still on the air (What is Murphy Brown?)
All 3 players bet everything and zeroed out. (Naomi said “the Bill Cosby Show”, Teri “Seinfeld”, and Jane “Married With Children”).
Because Jane was leading going into Final Jeopardy, the show gave Jane’s charity the $15,000 for first place (the U.S. Committee for UNICEF)
June 12, 1998 (Episode #3190)
This game is found in J! Archive.
Player #3, Marion Arkin, finished with $-400 and missed Final Jeopardy. Robert Levy and Steve Sosnick finished the Double Jeopardy! round tied at $7,600.
Final Jeopardy! category: ORGANIZATIONS
Final Jeopardy! clue: This women’s organization founded in 1890 was chartered by Congress in 1896 (the Daughters of the American Revolution)
Both players guessed the League of Women Voters and wagered everything. Three new players played the next Monday.
February 7, 2013 (Episode #6539, Teen Tournament Semifinal #2)
This game is found in J! Archive.
Scores going into Final Jeopardy:
Kelton Ellis $16,400
Joe Vertnike $12,000
Tori Amos $1,600
Final Jeopardy! category: CAPITAL CITIES
Final Jeopardy! clue: It’s criss-crossed by dozens of “peace walls” that separate its Catholic & Protestant neighborhoods (What is Belfast?)
All 3 players guessed Dublin and zeroed out. As per the rules at the time, a wildcard to the finals was opened. However, due to the inherent unfairness of this, the rules have since changed, as reported by ToC players recently. A tournament semifinal with three players finishing at $0 will still produce a finalist. It is believed that the current rules would settle this with a tiebreaker clue.
January 18, 2016 (Episode #7216)
This game is found in J! Archive.
Scores going into Final Jeopardy:
Mike Drummond $13,800
Claudia Corriere $13,800
Randi Kristensen $6,000
Final Jeopardy! category: STATE CAPITALS
Final Jeopardy! clue: A 1957 event led to the creation of a Natl. Historic Site in this city, signed by a pres. whose library is now there too (What is Little Rock, Arkansas?)
All three players bet everything and zeroed out. (Randi guessed Springfield, Claudia Austin, and Mike Atlanta.)
Update history
2016-01-09 original version
2016-01-10 updated with information on tournament semifinals, updated with information on 4/8/85 Final Jeopardy per Stefan Goodreau, updated to add correct response on 2/7/13 Final Jeopardy per David Kendall
2016-01-19 updated with 1/18/16 game as well as 3/2/98 game per Matt Carberry
2022-12-23 updated with more information about 4/8/85 per National Archives of Game Show History, more information on 7/9/91, reformatted page