Why, because it's part of the on-duty uniform.
The helmet isn't a hat and it doesn't just protect the head, it has many functions including communication and sensors.
From the Wikia
Its reinforced combat helmet featured an integrated comlink, audio pick-up, two artificial air-supply hoses, and a broadband communications antenna powered by a single power cell.
The helmet featured built-in filtration systems that extracted breathable atmosphere from polluted environments.
The helmet's visual processor assisted the wearer in seeing in darkness, glare, and smoke, though it limited the wearer's field of vision.
When firing a blaster, the helmet's visor polarized against the glare. A built-in heads-up display also provided targeting diagnostics, power levels and environmental readings at the corner of the wearer's eyesight, and one could access data on various military subjects and civilian organizations on the helmets display.
In addition, motion sensors alerted the wearer to any enemy the soldier might have missed.
Seeking to discourage nonessential chatter which was strictly off-limits while on-duty, stormtrooper helmets recorded everything that was said by the user, sending it to monitors to review after downloading the data off of the armor's memory.
As for when the Trooper is asleep...there is no reason for them to do so as they would be off-duty.
