Should I say
Do you know where Liz lives?
Do you know where does Liz live?
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Everyone else is giving the right answer: Can you tell me where Liz lives? is correct and Can you tell me where does Liz live? is incorrect. I will try to give a more intuitive explanation for why because it sounds like that it what you really want to know.
Let's start with questions and sentence structure. In English, word order is all we have, so let's pay special attention to a few examples.
Where does Liz live? is a full sentence that makes sense on its own. In fact, it must either stand on its own or be in a position that is appropriate for such an "independent clause". Please note that the fact that it has a verb, "does", is very important to whether or not it can stand on its own.
where Liz lives is not an independent clause. It is just a noun phrase -- you can think of it as a thing. It needs a verb and some other context to express a complete thought. For example: I like where Liz lives or I know where Liz lives.
Do you know ...? What is this question missing? This question is missing something to be asking about. Let me repeat: something to asking about, such as where Liz lives. Note: Where does Liz live is a fully independent clause and for this reason it does not fit after do you know.
Actually, let's take one more look at #3. There's two verbs in the sentence: "do" and "know". If we ignore the "do", the rest of the sentence can read you know where Liz lives. This is an independent clause and it if you said it by itself, it's not a question, but a fact. So when you are phrasing a question, "do" wants to be followed by an independent clause. What "do" does turn a fact into a question for you.
You should use the first one:
Do you know where Liz lives?
You might see something similar to your second construction if it's worded (and punctuated) a little differently. For example:
Can you tell me: Where does Liz live?
What was that guy asking you about just now?
Oh, he wanted to know: "Where does Liz live?"
The 2nd construction it totally incorrect. The first one is called "indirect question" which is a regular question with some introductory phrase.
Introductory phase can me anything, like
Do you know...
Can you tell me...
I am nor sure if...
Anything that will lead you into the question.
And after the introductory phrase, it is incorrect to use construction for questions (auxiliary verb + noun + main verb -> do you know, for ex.). Instead it comes positive.
Look at these examples.
What time is it? (aux + noun + main verb)
Where is the station? ( -||-)
Do you know... emphasized text
Can you tell me...what time it is
I wonder...what time it is
So in your case, the only correct option is to ask:
Where does Lize live?
Do you know where Liz lives? (if you want to be polite and make introduction to your question and not struck the other side with a direct question)