SHUT 字問題

You can say "shut the window/door" but not on air-conditioning or fan etc.
You say "turn off the air-con/fan."
However, I think you can say "shut down the air-conditioning system" if the system involves a network of air-conditioning units, or whatever.

"Shut the [expletive] up" means to cease talking.
If you talk to someone without stopping, or if you say nonsense to someone which upsets the receipient, that person will yell "shut the [expletive] up" to you.

只可以是shut the window,
door 的話是 : close the door,
air-condition and fan 便是: switch off the air-conditioner / fan.