Actually proper breathing technique plays a great role in exercise. In all cardio exercise one must know the art of breathing in and out. But in case of Yoga it plays the most important role. If anybody masters the art of Breathing by practicing Pranayama which is a very essential component of Yoga, he or she can perform any other workout in a more perfect fashion. Wrong breathing may actually harm you, whatever excercise you do.
Inviting your experience on the subject to enrich myself regarding the role of breathing in exercise
"Fire breath" has never been my cup of tea. I've done various forms of yoga. Some are more focused on breath than others. I have generally favored Hatha forms for the sense of flow and involvement of the whole body in movement.
Whether in martial arts, many forms of yoga, jogging, or any other exercise, I find it is generally beneficial to slow breathing down. Deep, slow breaths are helpful in several ways.