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.