CRT-tube-based televisions use a TV system comprising a scanning electron beam that paints energy onto the pixels of a phosphor-covered screen. The scanning pattern uses interlacing to draw each frame twice, on alternate lines, in order to make moving images appear to have higher resolution, among other things.
This scanning pattern shows up as annoying flicker when you shoot it with a video camera. Traditionally, the way to make that artifact go away is to adjust the electronic shutter in the camera until it synchronizes with the electron beam scanning pattern. Sony’s name for this camera feature is called “Clear Scan”, and comprises a knob or menu buttons that let you adjust the camera shutter in very small increments until the visual artifact is minimized or eliminated. Other brands offer this too but may call it something different.
If your TV screens are LCD or LED, this whole issue generally becomes moot. Those TV’s use a different method to address each pixel on the screen, instead of a scanning beam. However, clear scan “might” also be useful during 24-P shooting, where LED-based lighting uses a fast flicker rate as a dimmer, same for some fluorescent lighting.
Bottom line: check that the cameras you will use have clear-scan, or something like it, before the shoot. If shooting actual film, the TV’s on a movie set usually have their driving circuitry custom-modified to match to the crystal synch of the film camera, in effect putting the clear-scan device into the TV, instead of the camera.
There are specialists out there who adapt modern flatscreens into antique TV cabinets. Another method that’s more work in post would be to place 3 or more tracking markers on the old TV screen, and just have the TV tuned to static, so that it emits some practical light into the room, but then just use compositing software to insert the video of choice later, in post-production. The compositing route has to not just take into account the shape of the screen bezel, but also the slight distortion from the bulging curve of the front of the picture tube, and any internal reflections if there’s a flat protective glass over the front of that tube.
If the shot has no camera movement and is fixed at one angle, you might get away with feeding a green or blue signal from a signal generator, switcher, or idle DVD player, to the old TV, and generating a chromakey matte, but the replacement image will still need to be warped and positioned a little to match the screen angle. And too bright of a color projection creates spill into the room, which is a mess to clean up in post. So I would advise a gray or white signal and the tracking marker method over green or blue screen.