How do I use the auxiliary "do" in questions?

Consider these two sentences:
Who left the door open?
and
Who do you want to speak to?
In the first question auxiliary "do" cannot normally be used, but in the latter question it can. Why is that so? What is it about the second sentence that requires auxiliary "do" that the first sentence doesn't have? Please include a few more examples of this pattern if possible.

These are called subject questions (as your first example) and object questions (as your second). Here we are talking about "do", so this is present simple and past simple tense. Let us take one example:

Paul wants to speak to him.

1) If we want to know who "him" is, "him" is the object and the question we will ask is an object question. There will be an auxiliary verb, which will come between the question word and the subject:

Who does Paul want to speak to?

2) If we ask a question about "Paul" (the subject), however, it will be a subject question, and then all we need to do is substitute the subject with a question word:
Who wants to speak to him?
Further reading with examples: in this grammar book

3) It is possible to add an auxiliary verb in subject questions, that has been discussed here. To sum it up, it adds emphasis, as in:
A: Who wants to speak to him? Jack? 
B: No. 
A: Jim? 
B: No. 
A: Well, then who does want to speak to him? 
More examples of that can be found in the link above.