According to Major League Baseball an official no-hitter consists of a pitcher having pitched at least 9 innings while allowing 0 hits.
In MLB history has a team lost while having a pitcher throw an official no-hitter? If so what were the team(s)?
The only starting pitcher to lose a complete-game no-hitter was Ken Johnson of the Houston Colt 45s (pre-Astros), who lost to Cincinnati 1-0 on April 23, 1964. Johnson's throwing error allowed Pete Rose to get to second with one out in the top of the ninth. Chico Ruiz moved Rose to third on a ground ball, and second baseman Nellie Fox's error on a grounder by Vada Pinson let Rose score the game's only run.
The same page also notes a loss for a combined no-hitter:
Steve Barber (8 2/3 innings) and Stu Miller (1/3 inning) of the Baltimore Orioles lost a no-hitter to Detroit 2-1 on April 30, 1967. The Tigers got both their runs when Barber walked Norm Cash and Ray Oyler to start the top of the ninth. Earl Wilson (who started for Detroit and got the win) bunted them over. After Willie Horton popped up, Barber threw a wild pitch, which let Dick Tracewski (running for Cash) score to make it 1-1. Miller came in to pitch to Don Wert, whose ground ball was booted by shortstop Mark Belanger, allowing Jake Wood (running for Oyler) to score the go-ahead run.
By your question's definition of a no-hitter, the answer would be one game - Ken Johnson's start for the Houston Colt 45s against the Cincinnati Reds.
There have also been four 8 inning efforts that ended short for a full no-hitter due to the fact that the pitcher's team was losing. These were:
It should also be noted that there have been situations where 9 innings of no-hit baseball were pitched, yet the game went into extra innings, where the pitcher ended up earning a loss. Your question is mildly flawed, since it is quoting the original rule. The rule was changed in 1991 to not include games where 9 inning no-hitters were pitched, yet the game was still undecided (source). Other than the previously mentioned situations for Ken Johnson and Baltimore vs. Detroit, there are a few others of note:
Harvey Haddix of the St. Louis Cardinals pitched 12 innings of a perfect game in 1959 against the Milwaukee Braves, before it was broken by a fielding error. He then pitched two more outs before giving up a home run, losing the game and ending his no-hitter. His 12 2/3 inning performance is sometimes attributed as the best performance by any pitcher in major league baseball history, especially considering the Braves' lineup at the time (cough Hank Aaron cough).
In 1917, the Reds' Fred Toney and the Cubs' Hippo Vaughn both pitched 9 innings of no-hit baseball. In the top of the tenth, the Reds scored a run on a few hits, and Toney came back in the bottom of the inning to complete his 10-inning complete game no hitter, whereas Vaughn dealt with his loss after pitching 9 1/3 innings of a no-hitter. This is the only time in major league history that there were no hits by either team in 9 innings of baseball.
Jim Maloney of the Reds pitched 10 innings of no-hit ball in 1965 against the New York Mets. Johnny Lewis hit a home run to lead off the eleventh, and Maloney lost the game. Incidentally, he also pitched 9 innings of a no-hitter two months later, except he won in the tenth inning (the first time a pitcher scored a no-hitter in more than 9 innings). Maloney has a very interesting no-hitter repertoire that I won't go into here.