During the performance of "The Love Song" in the Partridge Family's living room, the closeups of Keith (David Cassidy) were clearly recorded at a different time from the rest of the show. In the closeups, the front of his hair is cut very short, with the bangs hanging down over his forehead, but in the long shots of the same performance - and indeed in the rest of the episode - his bangs are several inches longer and slicked to the sides.
During the musical number, Keith's hair changes from shot to shot.
In the opening scene, when Shirley is telling Keith and Danny about Laurie's problem, Chris had run out of the garage and it was just the three of them. They do a close up of one of the characters and when the go back to showing the three of them, it's now four because Tracey has mysteriously appeared out of nowhere.
The kid's transistor radio is a receiver only not a transmitter so it wouldn't be responsible for creating music in Laurie's mouth. Hpwever, there really is a rare phenomenon of dental work picking up radio signals but it's caused by fillings not braces.