show sb. the door
to make it obvious that you do not want someone to be present and that they should leave赶走（某人）；请（某人）离开；给（某人）下逐客令
show sb. the door意思是“（因不受欢迎而）撵某人出去”、“对某人下逐客令”。show sb. to the door意思是“把某人送到门口”，不可将两者混淆。
“When I told my bank that I wanted to borrow $100,000 they showed me the door.我告诉银行我想借10万美元时，他们直接让我走人。
open the door

open the door大家都知道是开门的意思，加上介词to/for，就会衍生更多的意思！open the door to sth可以解释为向...开门、打开...的门，还可以表示为...创造机会。而加上for，意思比较类似。open the door for sb/sth意思是为...开门、提供机会/方便、创作可能。
“Research on genes should open the door to the new medical treatments.对遗传基因的研究将会为全新的医疗方法创作机会。
If the record is successful, it could open doors for my career.如果这个记录成功了，将会为我的事业提供机会。
close/shut the door on sth
to make it impossible for something to happen, especially a plan or a solution to a problem使（尤指计划或解决方案）成为不可能
“There are fears that this latest move might have closed the door on a peaceful solution.人们担心这一最新动向或许已关闭了和平解决问题的大门。

knock the door
如果和外国人说knock the door，人家怕是误以为你要抢劫了。knock the door其实是破门而入的意思。
正确的表达应该是knock on the door
“He knocked on the door and waited for someone to answer.他敲了敲门，等待有人回应。或者knock at the door 美式口语中常用来表示敲门声
“She heard a knock at the door, so she opened it to see who was there.她听见有人敲门，就把门打开看看是谁。