When Casey looks through one of the doors, they see a version of them in a different reality: this reality is from the original Teenage Mutant Ninja Turtles (1987) cartoon. The score of the episode as well changes momentarily to mimic that of the older show.
The Cheese Phone is a homage to the Batphone from Batman (1966), where there was a special phone used exclusively to contact Batman and Robin for emergencies. The pose Splinter makes after using the Phone is a homage to Adam West's heroic Batman poses.