あなたの後ろのドアをしめてください。
“Please close the door behind you.”
I instinctively interpreted the English sentence as “Please close the door after you entered/went through”, but the Japanese sentence sounds to me like “Please close the door that is (currently) behind you”.
Can the Japanese sentence also be understood in both ways or is it just an unfortunate translation?
It’s an unfortunate translation in this case. The Japanese sentence is as straight forward as it can be, there’s no way to interpret it that way, unless you have an actual scene where you see the character just walk in through the door before being told that.