When using an italicized word with punctuation, should you italicize the punctuation, too?

This is ultimately a style question, and style questions are the province of individual authors or of the publishing houses responsible for producing their work.

At the magazines and book publishers where I've worked, we generally followed the "primary system" recommended in Chicago Manual of Style, fifteenth edition (2003):

6.3 Punctuation and font: primary system. All punctuation marks should appear in the same font—roman or italic—as the main or surrounding text, except for punctuation that belongs to a title or an exclamation in a different font.

Rendered in accordance with this guideline, your example would appear this way:

"Do we have to call it a 'kombi'?"

As you can see, kombi is italicized, but the single open and close quotation marks, the question mark, and the double close quotation mark are not.

Chicago also presents (at 6.5) an alternative system in which a period, comma, colon, or semicolon immediately following an italicized word receives italic treatment along with the word preceding it. But that system wouldn't apply to the quotation marks or to the question mark in your example in any case.

The one exception to Chicago's guideline 6.3 that some of the magazines where I've worked introduced involved apostrophes following italicized words: Because the apostrophe often crashed into the final italicized letter of the preceding word, we set the apostrophe as italic, too. Thus:

Finding a spiritually uplifting message in The Trial's predominant mood of grim determinism isn't easy.

It is worth reiterating, though, that style decisions regarding punctuation are not matters of universal rule, but of individual or house preference. As in other such matters, the most important practical consideration is to maintain consistency across the book, article, or publication series.