Can Schrödinger's cat be filmed?

To preserve the quantum superposition, the cat must be completely isolated from its environment, which means it cannot interact with the photons necessary for the camera to function. In other words, the camera cannot observe the cat without a flash, which is tantamount to making an observation.

Yes, it is possible to put a movie camera in the box with the cat. But when you open the box and look at the movie, you will see the cat die or not die, but not both. And that will agree with what you see when you then look at the cat: it will be either dead or alive. The movie film will be entangled with the cat. By opening the box and looking, you randomly select one state or the other, and that's all you see is the selected state. Some call this "collapse of the wavefunction", and some call it "branching of the world". For most practical purposes it's the same thing.

Write $a$ and $d$ for the alive and dead states of the cat.

Write $A$ and $D$ for the "contains a picture of a live cat" and "contains a picture of a dead cat" states of the camera.

Write ${cal A}$ and ${cal D}$ for the "saw a live cat and a picture thereof" and "saw a dead cat and a picture thereof" states of your friend Jeeter.

The cat starts out in state $a+d$. Then the camera goes off. Then Jeeter looks in the box. Then you question Jeeter about what he saw.

Theory I: Before the camera goes off, the cat spontaneously jumps into either state $a$ or state $d$. Then the camera goes off, entering state $A$ or $D$ accordingly. Then Jeter peeks in the box and enters state ${cal A}$ or ${cal D}$ accordingly. Now the whole system is either in the state $aA{cal A}$ or $dD{cal D}$. You run into Jeter and question him. Depending on the system state, he responds either that the cat is alive or the cat is dead.

Theory II: The cat is still in state $a+d$ when the camera goes off, collapsing it into either state $a$ or state $d$. The camera enters state $A$ or $D$ accordingly. Then Jeter peeks in the box and enters state ${cal A}$ or ${cal D}$ accordingly. Now the whole system is either in the state $aA{cal A}$ or $dD{cal D}$. You run into Jeter and question him. Depending on the system state, he responds either that the cat is alive or the cat is dead.

Theory III: The cat is still in state $a+d$ when the camera goes off. There is no collapse, so the cat-plus-camera is now in state $aA+dD$. Jeter peeks in the box, collapsing the state to either $aA$ or $dD$, and he enters state ${cal A}$ or ${cal D}$ accordingly. Now the whole system is either in the state $aA{cal A}$ or $dD{cal D}$. You run into Jeter and question him. Depending on the system state, he responds either that the cat is alive or the cat is dead.

Theory IV: The cat is still in state $a+d$ when the camera goes off. There is no collapse, so the cat-plus-camera is now in state $aA+dD$. Jeter peeks in the box, but does not cause a collapse, so the cat-plus-camera-plus-Jeter is now in state $aA{cal A}+dD{cal D}$. You run into Jeter and question him, causing the system to collapse to either $aA{cal A}$ or $dD{cal D}$, whereupon Jeter tells you either that the cat is alive or the cat is dead accordingly.

All of these theories (and many many others with many many more steps along the way) have exactly the same testable implications, so there are no reasons other than aesthetic ones for preferring one theory to another. Different people's aesthetic senses might lead them to prefer different theories, but no observable phenomenon can prove any of them wrong.

The details of all this were worked out by John von Neumann and appear in his book on the Mathematical Foundations of Quantum Mechanics.