The number of years next to each team's name is the number of full seasons they've played without winning. In the case of hockey, I am counting the 2004-05 season, even though it was never played. I am also counting the baseball season about to wrap up. I am not, however, counting the NFL season now in progress.
Teams that have been around less than 20 years, but have never won a World Championship -- or, at least, have never done so in their current metropolitan area
Oklahoma City Thunder 3 (won 1 as Seattle SuperSonics)
Washington Nationals 7 (never won as Montreal Expos, either)
Charlotte Bobcats 7
Houston Texans 9
New Orleans Hornets 9 (never won as Charlotte Hornets, either)
Memphis Grizzlies 10 (never won as Vancouver Grizzlies, either)
Minnesota Wild 12
Columbus Blue Jackets 12
Tampa Bay Rays 13 (1 Pennant)
Tennessee Titans 14 (never won as Houston Oilers, either, 1 AFC Title)
Nashville Predators 14
Phoenix Coyotes 15 (never won as "old" Winnipeg Jets, either)
Carolina Panthers 16 (1 NFC Title)
Jacksonville Jaguars 16
Vancouver/Memphis Grizzlies 16
Toronto Raptors 16
Colorado Rockies 18 (1 NL Pennant)
Florida Panthers 18 (1 Conference Title)
Ottawa Senators 19 (1 Conference Title, "old" Senators last won 84 years ago, but not same franchise)
Teams that haven't won in 20 to 29 years
Minnesota Twins 20
Washington Redskins 20
San Jose Sharks 20 (never won)
Edmonton Oilers 21 (1 Finals since then)
Cincinnati Reds 21
Oakland Athletics 22 (1 Pennant since then)
Orlando Magic 22 (never won, 3 Conference Titles)
Calgary Flames 22 (1 Finals since then)
Minnesota Timberwolves 22 (never won)
Arizona Cardinals 23 (1 NFC Title, never won in St. Louis, either, but did in Chicago)
Los Angeles Dodgers 23
Charlotte/New Orleans Hornets 23 (never won)
New York Mets 25 (1 Pennant since then)
Chicago Bears 26 (1 NFC Title since then)
Kansas City Royals 26
Sacramento Kings 26 (never won in Kansas City or Cincinnati, but did in Rochester)
Detroit Tigers 27 (1 Pennant since then)
Los Angeles Clippers 27 (never won in San Diego or Buffalo, either)
Oakland Raiders 28 (1 AFC Title since then)
Philadelphia 76ers 28 (1 Conference Title since then)
New York Islanders 28 (1 Conference Title since then)
Baltimore Orioles 28
Teams that haven't won in 30 to 39 years
Utah Jazz 32 (2 Conference Titles, never won in New Orleans, either)
Seattle SuperSonics/Oklahoma City Thunder 32 (1 Conference Title sine then)
Pittsburgh Pirates 32
Washington Wizards 33 (as Washington Bullets, 1 Conference Title since then)
Portland Trail Blazers 34 (2 Conference Titles since then)
Seattle Mariners 34 (never won)
Seattle Seahawks 35 (never won, 1 AFC Title)
Philadelphia Flyers 36 (6 lost Stanley Cup Finals since then)
Golden State Warriors 36
New Orleans/Utah Jazz 37 (never won, 2 Conference Titles)
Washington Capitals 37 (never won, 1 Conference Title)
Miami Dolphins 38 (2 AFC Titles since then)
New York Knicks 38 (2 Conference Titles since then)
Texas Rangers 39 (2 Pennants, could win this time, never won as "new" Washington Senators, either)
Winnipeg Jets 39 (never won, "old Jets" won 3 WHA titles)
Teams that haven't won in 40 to 49 years
Milwaukee Bucks 40 (1 Conference Title since then)
Vancouver Canucks 41 (never won, 3 Conference Titles)
Milwaukee Brewers 41 (never won, 1 Pennant)
Cleveland Cavaliers 41 (never won, 1 Conference Title)
Buffalo Sabres 41 (never won)
Buffalo Braves/San Diego/Los Angeles Clippers 41 (never won)
Kansas City Chiefs 42
San Diego Padres 42 (never won, 2 Pennants)
New York Jets 43
Atlanta Hawks 43 (never won, won 1 as St. Louis Hawks)
Cincinnati Bengals 43 (never won, 2 AFC Titles)
Phoenix Suns 43 (never won, 2 Conference Titles)
Indiana Pacers 44 (won 3 ABA Titles, 1 NBA Conference Title)
New Jersey Nets 44 (won 2 ABA Titles, 2 NBA Conference Titles)
St. Louis Blues 44 (never won, 3 Conference Titles)
Los Angeles Kings 44 (never won, 1 Conference Title)
Toronto Maple Leafs 44
Denver Nuggets 44 (never won)
Atlanta Falcons 45 (never won, 1 NFC Title)
Cleveland Browns 47
Houston Astros 49 (never won, 1 Pennant)
Teams that haven't won in over 50 years
Minnesota Vikings 50 (never won, 4 NFC Titles)
Washington Senators/Texas Rangers 51 (never won, 2 Pennants, could win this time)
Philadelphia Eagles 51 (2 NFC Titles since then)
Buffalo Bills 51 (never won, 2 AFL Titles)
Houston Oilers/Tennessee Titans 51 (never won, 2 AFL & 1 AFC Titles)
San Diego Chargers 51 (never won, 1 AFL & 1 AFC Title)
St. Louis/Atlanta Hawks 53
Detroit Lions 54
Rochester Royals/Sacramento Kings 60
Cleveland Indians 63 (3 Pennants since then)
Chicago/Arizona Cardinals 64 (2 Title Game appearances since then)
Chicago Cubs 103 (7 lost World Series since then, none of those in 66 years)
The Montreal Expos/Washington Nationals, the Charlotte Bobcats, the Houston Texans, the Charlotte/New Orleans Hornets, the Vancouver/Memphis Grizzlies, the Minnesota Wild, the Columbus Blue Jackets, the Nashville Predators, the Phoenix Coyotes, the Jacksonville Jaguars and the Toronto Raptors have never been to their sport's finals. Nor have the Oklahoma City Thunder in their current city.
The Bobcats, Texans, Hornets, Grizzlies, Blue Jackets, Predators, Coyotes, Raptors and (in their current city) Thunder have not reached their sport's final four.
The Texans (though they may do so this season) and the Nationals (since moving from Montreal) have never even made the Playoffs.
The Minnesota Twins, Cincinnati Reds, Oakland Athletics, Los Angeles Dodgers, Baltimore Orioles and Milwaukee Brewers have all gone at least 20 years, but not yet 30, without at least winning a Pennant. All but the Orioles and Reds have reached at least 1 LCS since 1997.
The Washington Redskins, Cincinnati Bengals and Miami Dolphins have all gone at least 20 years, but not yet 30, without at least reaching a Super Bowl. The Dolphins have at least reached a Conference Championship Game in that span.
The Portland Trail Blazers, Minnesota Timberwolves, Charlotte/New Orleans Hornets, Sacramento Kings and Los Angeles Clippers have all gone at least 20 years, but not yet 30, without at least reaching an NBA Finals. The Blazers, T-Wolves and Kings have at least reached a Conference Finals.
The San Jose Sharks and New York Islanders have gone at least 20 years, but not yet 30, without at least reaching a Stanley Cup Finals. Each has at least reached a Conference Finals.
The Pittsburgh Pirates, Washington Bullets/Wizards, Seattle Mariners, Golden State Warriors, Milwaukee Bucks, and (if you combine both franchises that have held the name) Winnipeg Jets have gone at least 30 years, but not yet 40, without at least reaching their sport's finals. The Pirates, Mariners and Bucks have at least reached their sport's final four, and the old Jets won 3 WHA Titles, but have never reached even the Conference Finals in the NHL.
The Buffalo Braves/San Diego/Los Angeles Clippers, St. Louis Blues, Kansas City Chiefs, New York Jets, St. Louis/Atlanta Hawks, Toronto Maple Leafs, Denver Nuggets and Cleveland Browns have gone at least 40 years, but not yet 50, without at least reaching their sport's finals. The Blues, Chiefs, Jets, Leafs and Browns have at least reached their sport's final four at least once. The Clippers have never won a single Playoff series since their franchise's creation.
The St. Louis/Atlanta Hawks, Detroit Lions, Rochester Royals/Cincinnati Royals/Kansas City Kings/Sacramento Kings and Chicago Cubs haven't reached their sport's finals in over 50 years -- the Royals/Kings and Cubs for over 60. In that time, final four appearances have happened to the Cubs three times, the Hawks and Lions twice each, and the Kings just once.
The city with at least 4 teams that has gone the longest without winning a title is Minneapolis-St. Paul, 20 years since the 1991 Twins. They are also the city with at least 4, or even at least 3, that's gone the longest without even reaching a Final. The city with at least 3 teams that has gone the longest without winning a title is Cleveland, 47 years since the 1964 Browns -- unless you combine Memphis and Nashville into one Tennessee region, which has 3 teams but has never won a title.
Of all cities with at least 2 teams, the ones that have never won have been Nashville (eligible since 1997), Buffalo (eligible since 1966) and San Diego (ditto).