LCD monitors don't really suffer from the same burn-in issues found on phosphor based display systems. However, virtually all LCD technologies are prone to a similar issue known as Temporary Image Retention (TIR). The good news is that panel technology has evolved rapidly and that TIR is unlikely to occur on more recent LCD panel designs. You would really have to try and cause TIR, perhaps by leaving a static 4:3 white box on a 16:9 panel for months at a time, but even the TIR you might see after doing so would not be permanent.
Simply turning the monitor off once and a while would help mitigate any potential TIR, but unless you are running a static image on the display 24/7/365 I would not worry about it. The greatest risk for TIR is using a 16:9 monitor only for 4:3 content in a 24hour per day environment for months at a time and then suddenly switching to 16:9 sources. In that scenario a vast majority of LCD panels will exhibit TIR, but again simply turning the unit off every once and a while or viewing 16:9 content every so often would help prevent this issue.
The important thing to remember is that phosphor based systems are much more prone to 'burn-in' than modern day high quality LCD panels are prone to 'TIR'. So even a fraction of the effort you would spend on preventing 'burn-in' on CRTs, Plasmas, etc. would suffice for preventing TIR on LCD panels.