LED lights do not “burn out” in a classic sense the way incandescent lights do but they can go out.
Most of the time, folks ask this question because they are concerned with the underlying question: If a bulb burns out the others will go out?
LED bulbs will go out if the semiconductor circuitry of the Light-Emitting-Diode is damaged. As long as the wiring harness (the Christmas light wires) is undamaged, the rest of the lights should be unaffected.
That being said, we must think about what would cause an LED bulb to go out in the first place. A manufacturing issue with the semiconductor component of a light string will show up within a handful of hours from the time the set is plugged in.
LED bulbs are most likely to be damaged by electrical surges on the electrical line. Created by lightning strikes, improperly used generators, hydroelectric or geothermal electricity generation, impulses are spikes of current on the electrical lines. These spikes of surges are deadly for LED lights in the same way that they killed our desktop computers in the 80’s.
Using a surge protector is the best and easiest way to filter out those spikes. Be sure to select an outdoor rated surge protector if your installation is outside.