In the days and weeks leading up to the burn you may start hearing about how alkaline the dust is and that your hair or skin may dry out in these harsh weather conditions. Most burners after their first burn create a plan leading up to the burn so they can limit their exposure once on playa.
I have not done all of the items that I am suggesting below. Instead I have rounded up what my community as a whole seems to do every year:
Hair: If the length of your hair is below your shoulders you should have a strategy for how you are going to manage it once on playa. Here's what happens: The wind whips up while you're biking and suddenly you have a tangle of dry, dirty hair to comb out. Many burners I know will go to their local salon and get braids right before they leave town. The braids can last several weeks and most of my friends will include strings of ribbons or bright faux hair extensions to create a colorful look. If you don't want to do braids (like me), you can opt to put your hair in a side or french braid during the day, and then wear it loose at night when it is less windy. You can also wear wigs at night to keep your hair clean and protected. If you have access to a shower I recommend washing your hair every day, or at least using a spray in conditioner and brushing it once a day.
Nails: The playa is notorious for wrecking nail beds and so a number of burners will get gels before the burn to protect their nails. Others will paint their nails, but I have never understood this since the polish only lasts a day or two. Other people will line their nail beds with super glue to protect them from the elements. If gels are too much I recommend washing your hands very thoroughly, once in the morning, and once in the evening before applying cuticle cream. After your hands have dried use liquid bandage to seal your cuticles.
Lashes: There's always a back and forth on whether to get lash extensions, and as someone who has done them every year, I can't recommend them enough. They easily stay on the entire burn, it alleviates the need to do makeup before going out, and you always look done and ready to go. If you decide to get lash extensions, just remember that you need oil free face wipes, otherwise you will degrade the glue. If lashes are too much of a commitment then consider from fun, funky fake lashes.
Wax: Shaving at the burn can be a pain, so if you prefer to not deal with the whole ordeal schedule a wax a few days before you head to Black Rock Desert. If body hair doesn't phase you, then skip this suggestion.
Skin Care: It's incredibly important to hydrate in the weeks before the burn. Your skin (and internal organs) will all be dehydrated by the time you leave, so give your skin a chance. I also recommend using lotion or coconut oil, even if it isn't part of your daily routine.
See my other suggested items for personal care on the playa.