Yes, it is possible to use a standard liquid crystal display to switch the polarization state of light. But, it may be impractical to do so, especially with modern thin LCD construction.
LCDs make use of two polarizing filters, 90 degrees to each other. The wikipedia article for LCDs has images and explanations of this. In principle, you could disassemble the display, remove the backlight (or reflective component as in a calculator display), remove the 'front' polarizer that faces the user entirely, find the orientation of the 'rear' polarizer that faced the backlight, orient your own polarized source to it, and control the LCD to change the polarization state.
In practice, disassembling an LCD like this may be quite difficult. Especially in modern LCD construction, such as what you find on mobile devices, since the polarizing films are often bonded to the glass substrate holding the LC and are not trivial to remove. Also, controlling the display could become difficult if you harm electrical components during disassembly.
For a more practical solution, there are products you can buy that use LC to switch polarization states. The link is just one example; perhaps there are better options out there.