Burn-in isn't a light being emitted when off, it's a light being dimmer when on.

An LED works by passing current between two different semiconductors. When an electron jumps over the "gap" between those two semiconductors, it releases a photon of a particular color (determined by the size of the gap). But over time, as an LED is used, the gap can be damaged (by heat, by vibration, etc); when this happens, fewer electrons can jump the gap and thus fewer electrons produce photons. Or the properties of the gap are changed so that they emit a slightly different wavelength photon. So if you leave a particular set of pixels on, producing light, for an extended time, those LEDs will degrade more than the rest of the screen, leading that area to be discolored or dimmer. This is burn-in.

Most of the time, that's fine, because the LEDs on your screen experience wear in a more-or-less uniform pattern. Your phone is somewhat less susceptible to this, since (1) you tend to have your phone screen off most of the time, (2) there aren't as many persistent HUD elements even when it's on, as every app has its own configuration of controls and UI elements, and (3) you tend to replace a phone more often than a monitor. When you replace your phone, it's probably more-or-less evenly dimmer overall than it was when you bought it, but since you don't have anything to compare it to, you won't know; with burn-in, though, that comparison is right next to the burned-in pixels.

By contrast, a computer monitor will typically be on for 8+ hours at a time, and persistent display elements are a part of every major operating system. If you're not using the LEDs in a panel more-or-less evenly, you'll end up with a persistent image.