Usually, shutting down the ILS means shutting down the localizer and glideslope signals. There is more than one reason to 'shut down' the ILS.
The ILS is designed to shut down automatically on detection of fault. FOr example, if the primary transmitter (of glideslope) malfunctions, the system will automatically transfer to the standby transmitter. If that fails or if the standby transmitter also malfunctions, the system automatically will shut down, with ATC being automatically informed. Cat III ILS systems are required to shut down within 2 seconds of fault detection.
FAA carries out flight check operations on various navigational aids in airports and can require the ATC to shut down the ILS if error is more than limits.
ILS can be shut down because of equipment change, maintenance etc. Of course, if the power goes out, ILS goes out with it.
As per regulations, any shutdown of ILS (or any other navaid) is to be informed via NOTAM.