Why is the text set in italic?
If the text is being set in italic to show that the text is being quoted, I would argue that the quotation marks are a part of the quote—so they should be set in italic too (Some may argue the opposite, but my last argument trumps this anyway). If you are setting the text in italic to add emphasis then it depends on...
What exactly is being set in italic?
If you are setting a portion of the quote in italic, having the quotation marks italicised doesn't make sense. First of all, the quotation marks are separate from the content of the quote and secondly if you had one quotation mark set in italic (because your italics were adjacent to a quotation mark, for example*), you would have to set the other quotation mark in italic too—They are a distinct pair, so having one italic and one not would make no sense. Ergo, if you have anything less than the whole quote set in italic, the quotation marks are not set in italic.
*The logic here is flawed anyway.
The aesthetic effect on the text
This is the most important point. If you decide to set only the body of the quote in italic, you're likely to run in to problems. The designer of the font you are using spent a lot of time manually adjusting the kerning between characters, including quotation marks, they didn't however spend any time looking at how the various styles worked together (not at this level, anyway). Punctuation is designed to work with the font it is a part of, nothing else. (side note: bold/italic/regular etc are different fonts, they are part of a larger typeface, not font).
Take these examples. Only the body of the quote is set in italic here:
Notice the large space between the opening quotation mark and the 'I', and the almost collision of the closing quotation mark and the question mark. Do you want to manually kern all of these in your text? Probably not.
This is the whole quote, including quotation marks, set in italic:
All of this is my personal opinion. Various style guides and authorities on typography seem to give different opinions on italics and punctuation but it seems that italicising the punctuation is the most common approach. This is mostly talking about punctuation in general and not specifically about quotes, though. There are some related discussions on the Stack Exchange network: