I've never witnessed a B-way mistake (with the exception of Eponine's books dropping into the front row when I saw Les Mis) so I'll have to contribute personal stories. Two of my stories involve children's theater, but they were funny, so I thought I'd include them anyway...
When I was in 6th grade, I was in a production of Rumpelstiltskin (yeah, no idea how to spell it) as Rumpelstiltskin and they completely skipped the "guessing the name" scene, and I had no idea what to do (I was backstage) so they just ended the show and I led the little kids on-stage and the audience was very, very confused. The worst part was that both adults in the show were on-stage at the time, and neither thought to fix it.
I'm now employed with the same company, and we recently did a production which was similar to the Wizard of Oz (but Western style, it's an odd show.) Anyway, the Wiz was played by an 8 year-old, and she played another part as well. No one told her that she didn't have enough time to go change in the bathroom and instead had to keep her costume backstage. I usually stood with her during this part (she spoke from backstage before being revealed), and the adult who was on-stage came to talk to her through the curtain, but she wasn't there, so I whispered through the curtain that she wasn't there and the adult needed to ad-lib, so she had a good minute on-stage with nothing to do, and the kids on-stage had no idea what was going on.
I was doing lights for a production in my high school, and two characters were supposed to begin kissing but be interrupted. Well, of course, no one interrupted them, so they just had to continue kissing for the next minute (quite literally) until someone finally just yelled the other girl's line from the other side of the stage to get her attention.
Why do we play with fire?
Why do we run our fingers through the flame?
Why do we leave our hands on the stove, although we know we're in for some pain?
-tick...tick...BOOM!