The real story
In the US, the 1990 Americans with Disabilities Act requires lift doors to stay open long enough for someone on crutches or in a wheelchair to get in. So in many lifts the 'close door' button is wired up but overridden until that minimum time has passed, and some building owners program it to do nothing at all in normal service. The industry's own trade group has said riders can't make the doors close any faster.
The button still matters. In firefighter mode, a key switch hands the lift to the fire crew, the automatic door closing switches off, and the doors only close while someone holds that button. Engineers call buttons that comfort without controlling 'placebo buttons'. In 2004 the New York Times found that most of the city's crosswalk buttons no longer did anything either.