If the battery in the sensor has finally died then the light will NOT go off, time for new sensor. The BCM will output code showing it cannot read it due to no sending the signal, but you have to get into it to read that code, or use a scanner that reads chassis codes too.
Early sensors like my '11 Versa the light goes off with simply airing tire up high enough. Later cars though must spin the tires to get them to read, they are not powered up until centrifugal force turns them on, it makes battery last longer.
Most new sensors must be 'switched on' when installed then registered to the BCM, both things are done when the sensor gets installed, tireshops wave a wi-fi device over them to do it.