Ymisa is absolutely right.
However, you should note TheFury's reply. It was clearly written before your sample dialog was posted. If you were asking a simple question to determine the price of an article, "How much does it cost" would be 100% correct and "How much it cost" would be wrong.
The confusion here shows clearly why it is necessary to provide context, so we know how you want to use the phrase in question.
Well, no, that's not quite true, Ymisa. Once again, it all depends on context. Volver's dialogue is:
A :What a nice mirror !
B: I don't even know how much it cost.
If A and B are standing looking in the window of a shop at the mirror, you would be right that B would say "I don't even know how much it costs".
If, however, A was admiring the mirror hanging on the wall in B's home and asked how much it cost, B would say: "You know, I don't even know how much it cost - my wife bought it". It would be "cost" (past tense).
Of course it is possible in past tense, I did not say using "how much it cost" can never be possible. But I assumed the original dialogue was to be constructed using present tense since the alternative to "how much it cost" was given as "how much does it cost". In that case both are wrong and the s is necessary, as you would agree.