Got my hair relaxed yesterday & I got a quater sized ozzing burn on my ear. Tried bacitracin zinc but it really hurts and itches. How long til it heals?

Irritant dermatitis. I assume the hairdresser spilled some of the relaxing chemical on your ear - which gives you an idea of how strong that stuff is! Bacitracin is not going to get it done; a steroid cream, if you have one, might help somewhat, and hold a washcloth soaked in warm water to the area in the evening, perhaps while you're watching tv. If it's not markedly better in a few days, let your doctor take a look.
Throw away the bacit. Bacitracin is useless. It is also one of the most common things people may be allergic to. THROW IT AWAY. Water is good for this. The itching suggests an allergic reaction to bacitracin. STOP BACITRACIN. A cortisone ointment or vaseline may be soothing. If it is severe you may want to see a dermatologist.