|MadSci Network: Astronomy|
The simple answer is "No."
If you do a Google (or other search engine) search on "sunburn moonlight" you will find these Web sites at Earth and Sky and "moonlight" at Wikipedia that discuss this question.
Accordingly, moonlight is roughly 500,000 times less intense than direct sunlight, and thus has so much less sunburn-causing UV that there is no danger of "moonburn." If one assumes that a sunburn can be caused by 0.5 hours in direct sunlight, and simply scales that time by 500,000, you would have to be in the moonlight about 28.5 years without stopping to obtain the same burn.
Try the links in the MadSci Library for more information on Astronomy.