isnt 别关门 also correct?
It is correct. "Instead of saying "do not" with 不要 (bùyào), we can say "don't" a little more quickly and forcefully by using 别 (bié)."
Accepted now 2021年3月1日
Why do we need 要 here? Why not just 不关門?
‘不关门’ means ‘not to close the door’ and in this exercise you're asked to translate “Don't close the door.”, wchich translates to ‘不要关门’. ‘不要’ together mean “Don't...” in requests and orders.
I believe 别关门 is also correct.
「別把們關上」not accepted, repoted