An electronic device may have a hinge that allows the device to be flexed about a bend axis. A display may span the bend axis. To facilitate bending about the bend axis without damage when the display is cold, a portion of the display that overlaps the bend axis may be selectively heated. The portion of the display that overlaps the bend axis may be self-heated by illuminating pixels in the portion of the display that overlap the bend axis or may be heated using a heating element or other heating structure that provides heat to the portion of the display overlapping the bend axis. Control circuitry may engage a latching mechanism that prevents opening and closing of the electronic device when the temperature of the portion of the display that overlaps the bend axis is below a predetermined temperature.