Shirts are best air-dried on a sturdy wooden hanger—it protects your clothes from the heat and violence of a drier, so they last longer and stay brighter. But a cramped and dark closet isn’t a great place for wet clothes and a drying rack can take up space on the floor.
If you have one available (and I recommend one over a gym membership), hang your shirts from an Iron Gym or other door-based pull up bar. Open the windows in the rooms on either side to take advantage of cross-ventilation and your clothes will dry in a few hours. Save your fabrics and save some energy.