Whoever said "The Chicago Manual of Style (6.8) says that When my friends ask, "What do you want for your birthday?," I never know how to respond. is the correct form." was most likely mistaken.
To begin with, they are probably referring to the 15th edition, where section 6.8 addresses periods and commas inside quotation marks, rather than the current 16th edition, where section 6.8 addresses punctuation with URLs and e-mail addresses. However, 15th edition section 6.8 does not address question marks and I could find no example of question-mark followed by comma followed by closing quote in the 15th edition. In any case the 15th edition is out of date.
The current edition (the 16th), forbids the construction of question-mark followed by comma followed by closing quote with one very specific exception. In section 6.119, punctuation that is part of a title is treated as if it is not punctuation, so if the title ends with a question mark, it would still be followed by a comma. However, if what is being quoted is not a title, then the comma is dropped as in these examples from the 16th edition:
“What’s the rush?” she wondered. (section 6.10)
Is it worth the risk? he wondered. (section 6.67)
“Are you a doctor?” asked Mahmoud. (section 6.119)
See also the Chicago Manual of Style Online Q&A where they change/correct
Can you believe that I said, “When she says, ‘Do you know which fruit Jim likes best: apples, bananas, or oranges?,’ tell her this: ‘Actually, I once overheard Jim say, “I only eat pears!” ’.”?!
Can you believe that I said, “When she says, ‘Do you know which fruit Jim likes best—apples, bananas, or oranges?’ tell her that, actually, I once overheard Jim say that he only eats pears”?