"Turn the temperature down" (to make the room cooler)
"Turn the temperature up" (to make the room less cool)
"Turn up the air conditioner" tends to refer to the speed of the fan. The faster the fan, the more cool air it'll produce. The slower the fan the less cool air it will produce. For example,
"Turn down the air conditioner. It's way too cold in here."
"Turn up the air conditioner. It's way too hot in here."