Broken screen casing

LCD displays do not emit any radiation. This is the same as LCDs that are used on laptops and on LCD High Definition TVs but these are much larger screens.
LCD are liquid crystal display just like the display on your digital watch, but it uses three lots of color crystals (red, blue & green)where as the digital watch uses one black crystal. The DVD player also a backlight panel to provide light behind the liquid crystals so you can see the movie images.

There is no danger in unscrewing the case etc, just don't have it turned on when you do this, because you may cause an electrical short to some components, even if you do it won't harm you (the elecrical voltage is too low) but it is enough to possibly damage the unit.