Burning Man doesn't rely on the 911 system for any of its EMS issues. There are two large aid stations that are staffed by professional doctors, nurses, paramedics, etc. (people who are certified to be able to provide Basic Life Support services at a minimum), which are open 24/7 while the event is taking place, and before and after for all of the people who volunteer their time to construct and deconstruct the city.
On top of that, there is a fully functional Advanced Life Support facility (called Rampart) that is established in the center of the city and is meant to handle the very serious cases, of which there are of course a few every year. They have the ability to radio for air transport to Reno (the nearest large city with a real hospital) at a moment's notice. Rampart is airlock-sealed to keep the pervasive dust out and provide an environment that is as sterile as can be expected.
Emergency services are taken very, very seriously at Burning Man, and they have a history that goes back a long, long way.