Why does Kant consider lying wrong? What is wrong with Kant's ethics when applied to the murderer at the door example?

Kant considers lying wrong because it violates the categorical imperative. Kant’s criteria for deciding whether an act is morally right or wrong is to ask oneself whether, “the maxim of your will could always hold at the same time as a principle establishing universal law.”

 This question has several components. First, the “maxim of your will” refers to a general rule derived from a specific event. So, if I decide to lie on my employment time card and say that a particular assignment took me four hours when it took me only two hours, the maxim of that decision might be articulated something like, “it is okay to lie when it results in personal gain,” in this case, monetary gain. The second part is that this maxim or rule would become a “universal law,” that is, it would be morally appropriate for everyone to follow in all similar instances. So then everyone should lie when it results in personal gain. Well, now my employer is going to go out of business because suddenly everyone is doubling their hours on their time cards. Clearly most people would not want everyone to take lying so lightly. 

When the categorical imperative is applied to the murderer at the door example, it would appear at first blush that Kant should have created a caveat to allow lying to be morally acceptable in instances where it would save a life. Why then did Kant consider every instance of lying to be morally reprehensible? Why can’t we universalize the maxim that lying is okay if you are lying to save a life? Kant speaks against making maxims so situation-specific that they lose their applicability. For example, he doesn’t permit a maxim to say something like, “If your name is Louis, and you live on the third floor 2B apartment on State street in Brooklyn, then you can lie about being too sick to show up for jury duty.” (This description is random, if there is such a Louis living in such a location, it is a coincidence.) That is, we cannot use a specific maxim to except ourselves from a more generally held moral law. Thus the murderer at the door example is rather esoteric, and makes it difficult to compose a valid maxim from it.

A second reason that Kant claims that it is better to say that lying is wrong at all times is the “slippery slope” notion. If the man at the door explains that he is not going to kill your loved one, he is just going to chop off both her arms, is that a good enough reason to lie? What about if the man at the door tells you he is just going to beat your loved one to a pulp? Well, what if he isn’t going to do her any physical harm, he is just going to take her at knife point and force her to withdraw all of her money from the bank for him? The real question at that point becomes, “what doesn’t justify lying?”

As for what is wrong with Kant’s ethics on this point, I think that most philosophers would agree that Kant is pretty internally consistent with his argument, and so a rebuttal to his application of his ethics to lying may need to be a meta-level argument. You might come up with an entirely different reason that lying is not always wrong, and use that to address his ethics.

I've linked to an study guide about the categorical imperative. The surrounding material on that page is informative as well.