In the "Elves and the Shoemaker" episode of Muppets Classic Theater, there is a scene toward the end where Rizzo the Rat comes on stage to give advice to Kermit and Robin. Since Steve Whitmire was already playing Kermit in that scene, Dave Goelz operated Rizzo for those two lines, with the intention of Whitmire dubbing the voice in later. (This is standard practice for Muppet productions.) However, Goelz's imitation of Rizzo's voice was so hilariously bad that they didn't bother re-dubbing it, leaving Rizzo with a very odd voice for that brief scene.
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.