The playa can be a dangerous place and getting sick on the playa is absolutely miserable, so do all you can to stay healthy. The first place to start is by reading the Survival Guide, and then the Official Burning Man Health And Safety pages. Much of staying healthy is using common sense, but below I will discuss some other tips I have picked up through experience.
This is the absolute most important step to staying healthy at the burn. I discuss this in detail on my Water page.
If you are on any medications, I suggest bringing extras to the burn as they are not easily replaced. Pack the duplicates separately so you don't lose them all (such as lost luggage, theft, etc...). Locking your spares in the car may be a good idea. This also applies to eyeglasses and contact lenses.
Although the Porta Potties at Burning Man are generally clean and none too offensive as far as odor, if you do get sick then walking a block or two to get to them may be unbearable. When I came down with flu like symptoms in 2013 WAG bags saved my ass - literally. WAG bags are inexpensive self contained systems for human waste disposal - simply open them up and place inside a 5 gallon bucket for emergencies. The bags contain a gelling agent and deodorizer, and once used go back into their own heavy duty zip lock bag. They can be found at many sporting goods stores, and on Ebay.
I find the playa extremely rough on my voice and throat. Bringing some kind of sore throat spray is a good idea. Menthol cough drops are also good, and consuming lots of pickles and /or pickle juice helps as well. Note that this refers to commercial vinegar brined pickles, not salt brined fermented pickles.
At the very least bring some band aids and disinfectant with you. More would be better and few people ever lament having too good a first aid kit when they need something. I find that liquid bandage is great for cuts on the playa as it seals out the dust. A compression bandage is also handy in case of a sprain.
Should your situation require actual medical attention, there are Med stations at the 3:00 Plaza, 9:00 Plaza, and on the Esplanade near Center Camp. These are staffed by licensed professionals, and can deal with any situation.
I highly recommend purchasing travel insurance because 5 hour ambulance rides to the nearest hospital are expensive, and helicopter airlifts even moreso. A two week travel policy only costs about 30.00 and covers those costs and many others should you have an emergency.
If you can get to a med station that is the best way, but if not have someone find a Black Rock Ranger who can radio for help. If the situation is life threatening flag down any official vehicle.
Note: Black Rock Rangers are volunteers who are also burners. They generally dress in khaki when on shift and carry radios when strolling around. Don't confuse them with BLM Rangers who are law enforcement officers, carry guns, and are often patrolling in vehicles.
If your problem (physical or mental) is related to being in an altered state, then contact the Black Rock Rangers or Med Station directly - they will be a lot more helpful and seem a lot less threatening.
Freedom To Be Stupid
Unlike the default world, there are very few people that will try to stop you from doing stupid or dangerous things as long as it doesn't put others at risk. Luckily there are people there to help pick up the pieces after something goes wrong.
Read The Back Of Your Ticket
Because of all the snarkyness and pranking, many people do not take the fine print on the back of their ticket seriously - my neighbors in 2014 actually were shocked when I told them it is serious and not a joke.
Seriously, read all the fine print there - it is a legal disclaimer that you agree to by using the ticket. People do get severely injured and / or die at the burn. I personally have on two occasions fractured bones there - a fractured wrist in 2012 and a broken ankle in 2014. The back of the ticket is eye opening and not a joke.