Liquid crystal displays (LCD) contain mixtures of several organic molecules. They would only be potentially dangerous if the display was broken open and even then, the toxicity of these compounds is probably low unless actually injested. Also, realize that a typical liquid crystal display contains very little liquid crystalline material.
There may be some health benefits from using an LCD versus a CRT, a TV style monitor, in that LCDs tend to be easier to look at for prolonged periods of time; this has to do with the refresh rate an that LCDs don't tend to flicker.
LCDs and plasma displays work on very different principles. A plasma display uses the excitation of a gas to give pixels. Think of it as though you had an array of very small fluorescent light bulbs (this is very approximate, but gets across the basic idea). An LCD works as a light shutter, and is composed of some kind of a back-light (usually an LED (light emitting diode)), a polarizer (just like in polarized sunglasses), an electrode, a layer of liquid crystalline material, a second electrode, color filters and a second polarizer. The molecules of the liquid crystal are reoriented by the electrodes, and depending on the orientation of the liquid crystal, light can pass through the two polarizers or not. I am not familiar with the health effects of plasma displays, although I believe they tend to be more energy efficient than LCDs.