Has a Major League Pitcher ever lost a perfect game because of something unusual? By unusual, I don’t mean hits, walks, or hit batters; I mean such unusual happenings as a dropped third strike, catcher’s interference, or something like that.
Milt Pappas lost a perfect game with two out in the ninth on a disputed ball-four call. He got the next batter for the no-hitter.
Harvey Haddix lost a no-hitter on an unusual play. Haddix had pitched 12 perfect innings (retiring 36 batters in a row) for the Pirates against the Brewers that day. However, Pittsburgh failed to score, so the game went into extra innings. In the 13th, the first batter (Felix Mantilla) got on base by means of an error. So, the perfect game was gone, but he still had the no-hitter. Mantilla was sacrified to second and Hank Aaron was then intentionally walked. The next batter, Joe Adcock then hit a home run. However, Adcock passed Aaron on the basepaths, thus rendering him put out, and his home run taken away (and replaced with a single). However, Haddix lost the no-hitter, the shutout and the game itself.
It’s not exactly what you’re looking for, but it’s probably as close as you’re going to get.
Zev Steinhardt
It’s not that unusual, but it’s my biggest baseball thrill:
Terry Mulholland threw a no-hit, no-walk game in 1991. The only batter reached via a error (by 3B Charlie Hayes, who also made a great stab on a liner to end the game).
I was sitting third-base side, eight rows up. 
Well, there’s the case of Ernie Shore, who sort-of threw a perfect game, but not exactly.
Babe Ruth was a pitcher for the Red Sox in those days. He pitched to one batter, and walked him. Babe was furious at the umpire, argued with him vehemently, and shoved him. The umpire ejected Ruth from the game, so the Red Sox brought in Ernie Shore to replace him. Well, Shore picked off the man on first base, and then retired the next 26 batters.
Was this a perfect game? Technically, no, because a batter DID reach first base… but Ernie Shore never allowed anyone to reach base, so he DID pitch a perfect game, didn’t he?
People still argue that one.