The Chicago Manual of Style, which a lot of publishers use even though it's not intended for fiction, says it depends on why you want to italicize.
If your line of dialogue is remembered by a POV character rather than happening in the story's live time, then both the words spoken and the quotation marks are italicized.
Hadn't Biff said "Are you going to believe him?"
If your line of dialogue is imagined, it appears in italics without quotation marks, and unless it starts the sentence, the first word is not capitalized.
I could practically hear Biff's whined are you going to believe him?
If your line of dialogue is internal thought by the POV character, it's in italics, no quotation marks.
She filled out the application, ignoring Biff's Are you going to believe him? while she listed her last three addresses.
If the words are mouthed but not said out loud, the words are in italics.
I finally caught Loman's eye. Are you going to believe him? I mouthed.
And last, if it's recorded or synthesized speech (from the TV, radio, computer, answering machine, phone, etc.) it's a matter of house style, often italics without quotation marks.
On the flat-screen, Biff Loman held papers but clearly read from cue cards. Are you going to believe him?
In the end, though, it always comes down to house style. Some publishers are rigid and others don't really distinguish among the different subtleties that would require italicizing speech and their CMOS rules. As already said by others, house style rules over all, and your best bet is to select your own "rule" and stick to it religiously. If it's different from your publisher's style, they'll change it.
Maryn, real good at muddying the waters