I agree, I think the two sentences are in fact not the same. 'The door doesn't want to open' means it should open, for example it's unlocked but it's stuck... and it is worth continuing to try to open it. 'the door will not open' means that it won't open, it might be locked for example, and it might not be worth trying.
"the door won't open" would be more idiomatic. As I advance through the levels of Portuguese, I am surprised at the number of literal translations into English. In some cases the discussion was simply locked without any changes being made. I am a little disappointed by these shortcomings. Get native speakers to check!!$
It's perfectly idiomatic in English. "The door won't open" and "the door doesn't want to open" have slightly different connotations. A person who says "The door won't open" doesn't think it is possible to open the door under current conditions. A person who says "The door doesn't want to open" thinks the door could still be opened under current conditions with a little more effort or a new approach.