We see the Moon because sunlight reflects off the surface of the Moon toward the Earth. The Moon does not produce any of its own light. Therefore, if what we see when we look at the Moon is reflected sunlight, can you get a moonburn? The simple answer is no, you cannot.
|Moonlight...nothing more than reflected sunlight.|
So why is a moonburn not possible? The amount of light reflected off the surface of the Moon and received by Earth is MUCH, MUCH less than received directly from the Sun. The reflected light from the Moon is 500,000 LESS intense than light directly from the Sun. In addition, the Moon does not reflect UV light as well as visible light, so the amount of UV light reflected off the lunar surface is even less. There just isn't enough UV light reaching Earth's surface from the Moon to produce a moonburn.
There are many claims of people sleeping outside under a full moon for several hours and waking up with a "burn" on their skin. This is "burn" is NOT caused by the Moon. It is NOT a moonburn. There could be a number of explanations, especially if the person has sensitive skin. It could be a rash of some kind, but it is certainly NOT a moonburn.