So, how does one go about "looking after" LCD displays? Is it enough just looking at them, so they don't go black from feeling lonely?
This is interesting from a camera point of view too, as
@Brownie can agree to.
The key to preventing image or pixel persistence is to frequently run a program (for calculators or computers) or use any self-test function that exercises the pixels, and couple this with frequently turning on and off the device and thus the display. General use of the calculator/computer/camera will usually ensure this type of exercise of the display and keep these issues at bay!
The real problems are with displays that have been sitting unused for long stretches of time! If this is the case then you can experience a variety of issues like stuck pixels (always on), pixels not turning on (always off) or pixel leak (dark smudges covering larger areas with no discernable individual pixels in them)!
For stuck pixels, there are a variety of fixes that can help. The first and least intrusive is to use a program that exercises the pixels or to run any selftest function (like for instance for the HP Voyager series of calculators by holding the * key down when turning the calculator on) that turns on all pixels or segments on the display.
This method can sometimes also be used for dead pixels as well, so try it out and also run it several times to see if it helps, as sometimes it can take a long time for pixels to unstick or come back to life especially if they haven't been used for a while.
If these tips wont fix the issues you can try various pressure remedies. This should always be done with caution, since, if you press to vigorously, it can further damage the display, but if you are aware of the risk, this has helped me clear both stuck and dead pixels. The idea is that the liquid in the liquid-crystal may have shifted and is not getting full electrical contact, or some connection is slightly misaligned from its contact point, which can sometimes be fixed by massaging the components of the display. There are a variety of approaches to this, which include tapping, rubbing, and pressing on or around the dead/stuck pixels... carefully!
If the above wont work then IF you can open up the device and get access to the contact strip (generally a rubber "sandwich" with two isolating layers on either side of a conductive layer with a number of contact points) that connects the electronics on the circuit board to the LCD-display there is a chance that any poor connections can be fixed by running the tip of a hot soldering iron along the PCB or flexcable where it connects to the LCD in order to reflow the solderjoints! The most well known use of this last method is probably when fixing dead lines in Gameboy screens!
Lastly, the most dreaded phenomenon, the pixel leaks! When you get to the stage where there is no longer any clear pixels that are stuck/dead but rather smudges instead, there is generally nothing more that can be done to fix this, short of swapping out the display for a working one from another unit! The most common reason for this in older types of LCD displays are that the materials used to create the different layers of the screen turn brittle with time! This is even more likely if the screen is not used often as the electrical charge that runs through the display when it is used heats it up ever so slightly but enough to keep the materials flexible!
Also, in my experience, problems with pixel leaks seems to be more prevalent with segment displays (like for example Game & Watch screens, status displays on cameras, etc.) that has larger elements, than with pixelated displays like on the Pocket Computers...
Hence the recommendation to exercise your old electronics periodically to hopefully not end up with this issue in the first place!
I hope that these tidbits of wisdom can help someone out!

I myself have managed to revive quite a few calculator displays using the exercise and massage techniques mentioned above!
Just be patient! And... good luck!