Why is 对不起周末我们关门 wrong?
Although it is not wrong, I wonder if many people would say it this way. 关门 also means close down and it is a bad luck to say it. Normally shops would use 休息 to rest.
In Taiwan we'd often use 打烊 (dǎyáng) to indicate the times outside of the normal business hours, but it's also the case here that 关门 (關門) sounds as if the shop won't be reopening.
Suggestion for "close" is 关闭, but the question won't accept it.
Also, I do believe 对不起，周末我们关门 should be accepted.
起 (qǐ) = to rise/to be able to
as in 对不起 (duìbùqǐ) = not able to face (lit.) = sorry
We also saw qǐ as:
起床 (qǐ chuáng) = get up from bed