Does a Wallpaper You Never Change Burn In
Burn-in needs three things: an OLED panel, a bright fixed pattern, and time. A desktop supplies all three without being asked.
On an OLED screen each pixel makes its own light and ages by how much of it it has made. A picture that never moves ages its pixels unevenly, and after long enough the difference shows as a faint ghost on a plain background. LCDs do not work this way and are effectively out of the argument, which is why the answer depends first on what panel is in front of you.
The wallpaper is rarely the worst offender, though. A taskbar, a dock and a row of icons sit in exactly the same place all day at high contrast, while the picture behind them at least changes when a window opens. What raises the risk is brightness held high for hours, a high-contrast pattern with hard edges, and the same image left in place for months.
The countermeasures are undramatic: lower the brightness a notch, hide the bar that never moves, and let the wallpaper rotate every day or two so no edge stays over the same pixels. A softer picture without harsh borders is also, conveniently, the kind that is easier to work in front of. Rotating it is less about the panel than about not looking at the same thing for a season.