If you're starting out as a new nurse and interested in the burn population, you should go for it. I think you'll find that a burn ICU can offer exposure to far more than "just" burns. Assuming you're working in a verified burn center, not just a unit that has burn in the name, you'll probably serve as a regional if not multi-state referral center.
I recently left an adult and pediatric burn center located in a level I trauma hospital, and I can guarantee that you don't "have to be basically healthy to burn yourself." We would see multi trauma patients with burns, tons of concurrent substance abuse and mental illness, plus, as another poster has noted a whole host of prior medical problems that will be exacerbated by their burn injury acutely, and during the potentially 6-12 month stay for our big burns.
The hyperdynamic nature of burn injury can induce MI or arrythmias in the patient with prior cardiac disease, renal failure in susceptible patients, then there is rhabdo from electrical injuries or unrecognized compartment syndrome from transferring facilities, massive PE due to immobility, and of course sepsis, sepsis, sepsis. We would also treat a fair number of patients with rare non-burn exfoliating skin diseases like necrotizing fasciitis, SJS/TENS, calciphylaxis, erythrodermic psoriasis, and plenty of others.
Assuming you work at this type of high acuity place you'll get exposure to tons of drips (pain, sedation, pressors, insulin, paralytics, continuous antibiotics), bedside procedures (trachs, bronchs, PEGs, bedside endoscopy), plenty of CRRT, and potentially unique vent modes or ECMO for patients with inhalation injury if your unit does that sort of thing. Your assessment skills will be tested as you simultaneously care for patients across the age spectrum, attempt to manage pain in the paralyzed patient, or try to walk the line keeping your non-vented patient breathing but not suffering as you give huge doses of meds for wound care.
As other people have mentioned it's crucial that you try to get into that unit for a shadowing experience, clinical rotation, or preferably capstone. You need to make sure that you really enjoy working in a super heated room (burn patients get hypothermic easily) while dressed in a plastic wrapper, and that the sight and smell of huge wounds doesn't bother you. We have had both students and regular employees lose consciousness during wound care. Our unit would happily hire new grads, but they were required to spend time on the unit (preferably during a capstone) to make sure they could deal with the realities of burn ICU care.
There are plenty of people who have spent their entire career in burns-maybe you'll be one of them. One of the underappreciated benefits is your coworkers: people generally don't come to burns just because there was a position open. People come because they're specifically interested in the field, and want to do what they're doing. These are they types of people you want to work with.