I hate to contradict Su.Ki, whose thorough explanations are always as entertaining as they are educational, but I do think there is an instance where, in AmE, “the day is up” works. If you use “day” to mean “the period of daylight” then it is something that expires. “I’m hoping to get to my next campsite before the day is up. I don’t like hiking in the dark.”

No, you can't use 'up' in this way - it doesn't mean anything. You can use 'over' ( or occasionally, 'through') to indicate that the day or night is finished, but not 'up'. The only way that 'up' can mean 'over' in this sense is in a phrase such as 'Time is up!' or 'The two hours I had given myself to do my essay were soon up'. This use of 'up' suggests a notion of expiry (in that you've run out of time). You can use 'up' to refer to a specific length of time running out ( five minutes, two days, or simply 'time') , especially if it's the time allotted to something, such as a test or game. You can't use 'up' for the actual 'name' of the period ( a day, a night, a season. a year, a course, a test). For these nouns, you can use 'finished' or 'over'.

the day is up , actually, means the night is over. "To be up" means to be over or to finish or to end or to run out (like, the milk in the bottle is up).