How could light from a TV screen refract when viewed through a helmet's transparent visor? How could light from a TV screen refract when viewed through a helmet's  transparent visor?

 A: The TV display is an lcd display, its light is polarized.
The helmet's visor is birefringent because of strains in the polymer material. That will rotate the plane of polarization, different for different wavelengths.
The reflection is different for different polarizations, it acts as the analyzer here.
The colors are not really the colors of a rainbow, but complementary colors of the primary colors. Where the red from the display gets extinguished you will see cyan. Where the blue is extinguished you see yellow. Where the green is missing you see magenta.
This phenomenon is indeed not due to refraction.
Edit: I took your photo and tried to show the origin of the colors. In three strips, the RGB channels are shown. And then one sees that the cyan bands occur where the red intensity is lowest. Etc.

A: It is very simple to understand
For rainbow formation we need white light , water droplet, and air
Here LED/LCD screen are designed by polariser which act like water droplet and helmet lid is polymer which is another medium where light can enter.
So here we got our light--> TV's screen(droplet) -->helmet another refrective medium and here we got VIBGYOR
LED / LCD stand for light emitting diode and liquid crystal display which dispers the light you can see rainbow on tv screen by 30° angle
