Will displaying the same for a very long time on a graphical or pixel mapped LCD harm the display in any way?
If so, what would be a maximum time to display the same, or how could I determine that time? Would it be a good idea to implement some sort of screensaver?
The quick answer is, yes. But not in the same way as the old CRT's or Plasma screens.
The primary thing that degrades in any LCD is the backlight. Over the lifespan of the LCD, the backlight will get progressively dimmer. This is true of all backlights: LED, Cold Cathode Fluorescent, and Electro-luminescent. To slow down or prevent this degradation you can dim or turn off the backlight when not in use.
The next thing that degrades in LCD's is the LCD "material" itself. This happens to the Blue sub-pixels more quickly than the Red or Green, and happens because the light-energy shining through the LCD gets absorbed by the LCD itself and heats it up. For most "direct view" LCD's this is a non-issue. There just isn't enough light going through the LCD to do anything. But if you have a video projector that is shining a very intense light through the LCD then you need to consider this. Also, if your LCD is in direct sunlight for a lot of the time then you might have issues. The solution to this is to turn off or dim the backlight/projector-bulb.
What I cannot tell you is how important it is to do the backlight dimming. Some cheaper displays will have more problems than better quality displays. Without knowing specifics, I can't tell you specifics. I can tell you that I design equipment that is supposed to run for 10+ years (24 hours a day, 7 days a week), and we always dim or turn off the displays when not in use.
LCD displays use an external light source, and polarization of individual liquid crystal cells to allow / prevent light passing through. Pixels are not actually "lit", unlike in CRT or plasma displays. Thus, they really don't have any pixel element that could degrade with being on all the time.
As a matter of fact, individual LCD pixels are not "on" or "off", each is in one of two polarization states, both "on". You can check this by carefully prying off the uppermost polarization glass sheet of an old LCD module, then putting it back flipped over - the former "on" pixels will now appear "off".
Pixel death does occur, not because of specific pixels being in a given state continuously, but because of marginal manufacturing defects, or clean-room contamination. This would be seen, for instance, in DoA pixels on most LCD TVs or monitors, as well as on low cost graphical LCD modules.
This type of failure is not just "DoA" (dead on arrival), but could occur subsequently, either due to the marginal defects mentioned above deteriorating due to use, or due to contact oxidation at the connections to the LCD panel, over time. The actual on/off state of individual pixels has little or nothing to do with this.
OLED displays might conceivably degrade due to individual pixels being left on for extended periods, just as any conventional LED degrades and loses some luminosity over time, but from reading various publications, it seems safe to assume that the time to perceptible degradation is in decades.
On the other hand, the backlight is prone to failure due to extended use. Common backlight technologies such as CFL or electroluminescent (EL) panels deteriorate faster than LED backlights, but all have finite and relatively brief operating lives, years rather than decades.
Edit: I notice David Kessner has excellently addresed the backlight / sunlight issue in another answer.
Footnote: Solutions for the paranoid - invisible screensavers...
One method which I have heard of being used with televisions, though mostly through anecdotal evidence, is to shift the entire display by a random small number of pixels along each axis every so many hours. That way, individual pixels will be subjected to at least some relief, except within bodies of solid color within the display area.
The eye does not notice such shifts, yet the net result is similar to having a screensaver in place.
There is a third mechanism for the degradation of an LCD:
There is the anchoring layer, which is a thin layer of transparent material that is formed over the surface of the electrodes and onto which the ends of the Liquid crystals "anchor". This layer is often (but not always) a polyimide layer which has a "rubbing" to provide the correct orientation.
The principal mechanism for this failure of optical rotation is the exposure of the anchor layer to UV and the trapping of surface charge. This causes the LC to un-anchor and then subsequently it fails to rotate the polarization. The higher the energy of photons the greater the chance of degradation over time.
LED backlights (with three colors of LED's) will be safest as the UV content will be least. UV exposure from the front side will also adversely affect this layer as well. If you have an applied rotation and it's exposed to UV then this can be imprinted into the rubbing layer as a permanent effect. So a screen saver can help.
There are LC formulations that are insensitive to UV and there are anchoring systems that are better at UV exposure. most of which are trade secrets.
Another factor not yet mentioned is that while displaying a constant image on an LCD will not damage it, some LCDs may be damaged by images which flicker at a rate which coincides with the rate at which they switch drive polarity. It's possible to drive a pixel with positive or negative voltage, and the darkness of each pixel will depend upon the magnitude of the voltage rather than its polarity, but the average voltage with which each pixel is driven must be close to zero. Displays generally take care of this by flipping their drive polarity at some interval (which may or may not match the frame rate). If the display is showing a constant image, each pixel will be driven exactly as hard when the polarity is one was as when it is the other. If one were to turn a pixel "on" any time the display was driven with one polarity, and turn it "off" every time the display was driven with the opposite polarity, however, that could cause a polarity imbalance which could in the short term (over a period of seconds or minutes) cause ghosting effects that could take minutes or hours of balanced driving to dissipate; if the imbalance continued long enough, the damage could become permanent. Trying to get a 50% gray by turning the display on for one frame and off for one frame may seem workable, but could cause relatively rapid damage. A better approach for getting "grayscale" is to use a 3-frame pattern (on 1 off 2 or off 1 on 2). This avoids burn-in problems and also provides another gray level. Not sure why I've not seen display controllers offer it as a feature (even though some do offer a "gray" mode which is on 2 off 2).
LCD uses a charge to store the voltage which biases the contrast and brightness of the shutter effect of pixel. After 5 yrs or so, some screens develop a memory effect, which comes from ageing, where you can see the negative of a long-term steady image on the screen.
Turn the LCD off when not in use. It also extends the backlight life.
