How to reduce or eliminate LCD burn-in and stuck pixels

I have an old-but-still-nice 30" monitor on my desk, a Dell 3007WFP. This display has served me well for years, but recently, it's started having issues with "burn-in" or after-images created by displaying the same icons and windows in the same basic spot over time. I think the problem has been made worse by the fact that my system, for whatever reason, sometimes resists going into power-save mode and stays on for hours when I'm away from the computer.

The problem got to be bad enough recently that I was seriously contemplating retiring this monitor and replacing it with something newer. However, before doing so, I decided to try a software solution that claims to reduce burn-in and repair stuck pixels. Specifically, since this is a Windows PC, I ...