In the Candice Bergen episode of The Muppet Show, Kermit ended the episode by attempting to pie Fozzie Bear in the face. However, Kermit didn't get enough cream on Fozzie's face, so he pied Fozzie again and ended up hitting part of Bergen's shirt. Fozzie said, "Look what you just did!"
A change that became an establishing character moment for Miss Piggy: in one of her earliest appearances, the script called for her to slap Kermit over some disagreement. Frank Oz instead had her karate-chop Kermit with a wild yell. The rest is history.
More of a meta example, but it still fits. During preproduction for Muppets Tonight, the performers started fooling around with random characters to work out characters for the show. When Bill Baretta started singing with the Johnny Fiama puppet, no one was listening, so Brian Henson picked up a random monkey puppet and yelled at everyone to listen to Johnny. This resulted in the character of Sal.
Sesame Street is known to do this with their "Muppet and Kid" segments, in which the Muppet asks the kid something and they reply. In the most well-known of the bunch, Kermit was supposed to sing the ABC's with a little girl. Only the child decided to have her own little joke by shouting "Cookie Monster!" at various points in the song. Jim Henson played along with it, creating one of the funniest moments in the show's history.
A more dramatic example from Thunderbirds: while filming the climactic landing sequence of "Trapped In The Sky", the string holding one of the Elevator Cars onto the rolling road suddenly snapped, causing it to veer out of control and crash. Special effects director Derek Meddings was so impressed by the realism of the crash that he convinced the Andersons to rewrite the script to incorporate it, adding a shot of the out-of-control Elevator Car crashing into a parked aircraft.