LCD's don't burn but they can develop what is called persistence of image when they display the same thing over time. The strength or effect varies from manufacture to manufacture. However unlike CRT's or Plasma's, it is not permanent. Shutting the LCD down and unplugging for a day or so will clear up the image shadow. This may not be ideal for you.
You will have more luck with inexpensive LCD's designed for TV use as they are generally lower resolution than computer LCD's and larger because they are designed for distance viewing.