Continuity mistake: The space shuttle at Nasada - first it is below ground (as shown by the little amounts of light), then it is in the open (we see the Rangers get inside the shuttle, and it is a sunny day), and then we see the shuttle be raised up from the underground.
Continuity mistake: Only a minor mistake - when Japanese footage is used, the Rangers, Elgar, Rygog, and Porto look slightly different - for example, Elgar & co. look more menacing and detailed (compared to their American counterparts), whereas the Rangers costumes are made of a different material.
Continuity mistake: The Rangers try to use the Rescue Megazord's Artillery Power to defeat Divatox's monster. After it fails, the monster blasts energy at the Rescue Megazord - at which point its shoulder-mounted Artillery cannons disappear. They reappear again in the next shot.
Continuity mistake: One at a time, the Rangers jump down a tunnel into the Phantom Ranger's service tunnels. After TJ has jumped down, watch Carlos - his position changes between frames, and the camera is on him the whole time. First he is facing the hole, next he has his back turned - yet we do not see the movement.