My wife is a therapist. Most therapists get something called peer supervision, which is basically therapy for therapists. My wife sees about 20 clients a week and has one or two hours of supervision a week.
She also has to practice what she preaches. Mindfulness, meditation, sports. All the things you'd do to feel physically and mentally fit.
Excellent answers offered so far. I'll rephrase the question: what conditions lend themselves to burnout?
Overworking is one. I like to see about 25 to 30 clients per week; more than that is too much.
Being careful with regard to how many really needy/suicidal folk I take on at once is another, though it's difficult to regulate that.
Having colleagues to talk about the more challenging cases with is super helpful. I have a regular consultation group I participate with.
Administration overhead in a private practice contributes to burnout and needs to be regularly reviewed and minimized. I've stopped working with most insurance companies at this point because they underpay and offer too many restrictions and micro-management to be worth it.
You've got to keep the rest of your life in order so that other people's problems don't compound on top of your own. That means self-care and honoring and maintaining one's relationships. If I'm feeling loved and loving in my personal life I'm much more able to be a good therapeutic presence for my clients.
It helps tremendously to stay mindful with regard to what is happening in the therapy room. Some clients present as hopeless. Others get angry or upset with you. Others want so much to please you that they tell you they're doing well when that isn't the case. Still others can't seem to make progress even though they want to. It's so easy to take this sort of thing personally but that is a mistake. Its all just behavior you can reflect back to them so as to stimulate the client's own mindfulness. Of course, all the self-care and social support is what makes my own mindfulness possible.
This is the best answer, it encompasses almost everything. I would just add that my own therapy and then peer supervision/consultation are super helpful. I also find that you must have friends who aren't therapists (in addition to the ones who are). You need to be able to talk about "normal" things and not just work/therapy theory.