Note that I am talking about temporary image persistence here that disappears when the monitor has had a few hours rest. The monitor in question is a Samsung SyncMaster 245BW, which is probably about 8 years old, and been used pretty continuously. I’ve recently started noting a few such artifacts which go away when the monitor is switched off overnight. They’re weird, though. Specifically - I don’t tend to move or resize my browser much - I just close or minimize it when not using it, or open something else on top of it, and it probably uses about 3/4 of the screen width when opened, and most of the height. The artifact I really notice is a “ghost” of the border right underneath the tab area, plus an outline of the left most tab border. I find it curious that, given that, I have to look really close to see an artifact of the taskbar if I hide it (I utterly HATE the “Autohide task bar option”, so the bar is always there, unless I go out of my way to hide it). It didn’t used to do this - an excuse to buy a new monitor?
The LCD shutters can wear out, yes, producing slower-responding and unresponsive pixels.
But “persistence” in the sense of CRT or plasma burn, no. I’m not even sure that the wearout is due to continuous white/color value at that point or because of excessive level changes - literally worn out from changing phase, rather than stuck from always showing one value.
New, good, big monitors are cheap. Knock yerself out. 
The exact mechanism is not pinned down, but yes LCD shadowing can happen on some monitors. My mid tier 24" PC monitor with a static Windows taskbar is about 7 years old and has no shadowing at all so it’s not inevitable.
What exactly do you refer to when you say LCD “shutters” with respect to a direct view LCD screen?
Yeah, I know it happens. I just wondered if it was normal as a symptom of age - it’s reasonable that it would be. I was set to bump up from 24" to 28 or 30 anyway.
Note that OLED displays - more popular in phones than desktop displays - have a reverse burn-in problem. The different colors “wear out” at different rates, so you can end up with what’s effectively an inverted image.
I understand that there has been quite of a bit of improvement in the field since I was looking at them in my last life as a phone designer, so it might not be as big a deal any more.
Individual LCD cells. I might be using an antique term.