An electronic device may have a display cover layer provided with an infrared-transparent antireflection coating. A pixel array may emit visible light through the cover layer and the coating. An infrared emitter may emit infrared light and an infrared sensor may receive infrared light through the coating and the cover layer. The coating may include a stack of thin-film interference layers. The stack may include alternating lower and higher refractive index layers. The layers may have thicknesses and materials that configure the coating to exhibit an infrared transmittance of greater than 94% from 920 nm to 960 nm and a photopic reflectance of less than 1.5%. The coating may reflect visible light to prevent displayed images from being obscured by visible reflections. At the same time, some photopic reflectance of the coating may be sacrificed to maximize infrared transmittance and accommodate operation by the infrared emitter and sensor.