In the old days it used to be a common standard that you would use a screensaver to protect your display from image burning. Image burning is a phenomenon where your screen would stay still too long and the image in it would get burned to the screen permanently. Nowadays, image burning can certainly still happen but it is rare since the technology behind our screen has developed so much since the 1990’s.
In any case if you are concerned about your display here is what you can do to protect it from any damage.
1. Only have your computer screen on when you actually use the computer. If you have a laptop then only have your laptop on when you actually use it. I run my desktop computer pretty much 24/7 and when I do not use it I always turn off the display.
2. Use an actual screensaver. In Unix world there are plenty of them: xscreensaver and gnome-screensaver are two good examples. *See extra notes.
3, Third choice is to put xset -dpms s on & (or similar) to your system’s automatic startup scripts. Dpms will try to automatically shutdown your display after some time has passed without an action. I have found this step to be mixed results approach: with some computers it works and with others it does not. With PostX Gnu/Linux I have always set this option to off – since it has been troublesome with some laptops in particular.
4. Hibernate or suspend your computer. This is basically the same for the screen like turning the computer off – since the screen gets to shutdown.
In my opinion, the best option for you is to turn off your desktop computer’s display or turn off your laptop computer/desktop computer when you are not using it. Especially with laptops, turning off totally is a good thing since laptops do break more easily overtime than regular desktop computers.
Also, remember that whatever you do your display is going to break down some day. I have always been careful with my technology and just today my display is showing signs of getting totally broken soon. It has a purple vertical line in it and it has some dead pixels and even some image burning in it. This particular Samsung display served me well but few days ago it started getting really hot and showed some strange behavior, which was the beginning of the end, apparently.
My simple advice for displays and computers is this: If in doubt turn it off when you are not using it. With computers pretty much anything can break down and sometimes just being cautious might give your technology some extra days – if you are lucky.
Extra notes for point 2: In some very rare cases screensavers will actually do you any good these days. Screensavers were used in CRT monitors and the threat of image burning was very real back then. LCD and LED displays have made the technology that manifested as image burning pretty much a thing of the past. So, in theory screensavers could protect you – but in practice they are unlikely to be any kind of advantage these days.