I know he wears a hat on the cover of every book in the series. But, does it ever actually mention him wearing a hat in the books?
The publisher is responsible for the hat. The logic was evidently 'Wizard + P.I. = Duster, Staff, Fedora.' The fact that this contradicts the book is a non-issue for them - cover art is more a marketing item than anything else and the author has little to no say in it.
Jim has mentioned this in Q&A Sessions:
I INTRODUCED THE NOVELS TO MY FIANCEE WHO IMMEDIATELY CONSUMED ALL OF THEM [IN THE SPACE OF A WEEK?] AND AT THE END OF IT SHE LOOKED AT ME AND SHE SAID "I REALLY LOVE THESE NOVELS BUT - WHY IN THE COVER ART DOES HE WEAR A HAT?
What's with Dresden's hat in the cover art is the question - the answer to that is the art department thought it was a good idea. They thought it was the perfect visual shorthand for wizard detective - he's got a wizard staff and he's got a detective fedora - wizard/detective right there. So that's what they wanted and that's what they got. And they said - Jim, you approve of this -right?!?! [Laughter]
Word is the Dresden Files cover artist (Chris McGrath) has become a fan of the books after doing the first few covers, and understands that the hat is contradictory. At this point it's a bit of an in-joke for fans, and the line in Changes was a reference to the covers. Unfortunately the Jim Butcher forums where I read this has had an upgrade (or crash, or something) and none of the old topics that had quotes regarding this are still around. There are plenty of un-cited comments about it though.
Harry doesn't wear a hat in canon and even states that he doesn't like to wear them*, even though every cover so far has shown him with a hat. He's worn a hat for limited times when it has been required for the job (Fool Moon, Summer Knight, Proven Guilty), but doesn't regularly wear one.
*in Changes p.298 "A weight suddenly enclosed my head. I sighed patiently and reached up to remove a conquistador's helmet decorated to match the armor. I put it down on the floor of the limo and said firmly, "I don't do hats."
Also of note is that he did not wear a hat in the TV Series.
- Jim Butcher's Dresden Files - Comic by Dynamite. All illustrations show him sans hat.
- Backup, short story by Jim Butcher. His brother describes him with his iconic look, sans hat.
- TV Tropes defines the trope of Robe and Wizard Hat, and how Harry defies definition by this trope.
- Fred Hicks (creator of the RPG) posts on G+ about the miniatures and why Harry doesn't have a hat (and references the line in the books I can't currently find).
The artist once said, that he had drawn several drawings of Harry with hat and without, and the drawings with hat simply looked better.
In one of the short stories - the beer related one with Gard - Harry thinks about beginning wearing a hat.
The falling rain would wash away the spell on the amulet if I left it out in the open, so I shielded it as carefully as I could with the building and my hand. A hat would have come in handy for that purpose, actually. Maybe I should get one.
“I got your back,” I told him. I lifted a section of my long leather duster with my staff, so that I could hold the amulet in the shelter it offered. I looked only moderately ridiculous while doing so.
I’m going to get a hat one of these days. I swear.
— Short story "Heorot", originally published in My Big Fat Supernatural Honeymoon. Emphasis added
First book of the series, STORM FRONT. Chapter 3, 2nd page in.
"Marcone seemed somewhat put off by my attitude. Maybe I was supposed to be holding my hat in my hand, but I had never really liked Francis Ford Coppola, and I didn't have a Godfather."
This is the only reference to Dresden casually wearing a hat that I am aware of. But the book never stated that he put his hat on before he left his office. So either he was wearing a hat that was, and is never mentioned in any of the books. Or the hat is symbolic, mentioned only to further the reference to the Godfather movie.
When riding in the Leanan Shide's limo in Changes, Lea puts a Conquistador helmet on Harry. To which he responds by removing it and stating "I don't do hats". It's clear that Harry owns "hats", but they have all been used to complete a disguise of some sort as in Fool Moon when he uses the bland potion to hide from the police or in Summer Knight when he dons a delivery uniform to sneak past Ronald Ruel's door man. He has never been known to wear a hat casually.
After a quick search through the books the only thing I could find was the following.
Susan's mouth quirked in a malicious, predatory little smile, and she came in, folding down her umbrella and brushing past me. "This is your apartment?"
"Nah," I told her. "This is my summer home in Zurich." She eyed me as I closed the door, took her coat, and hung it up on a tall old wooden hat stand near the doorway.
So we know that he has a hat stand in his apartment which would indicate that he has a hat or at least had one at some point. Also we know from the book Summer Knight that he uses hats as a disguise.
I walked in wearing a hat with an FTD logo and carrying a long white flower box under one arm. I nodded to an aging security guard at a desk and went on past him to the stairs, my steps purposeful. You'd be surprised how far a hat, a box, and a confident stride can get you.
From all this and the covers as you mentioned I would say it’s pretty certain that yes he wears a hat.
HE DOESN'T WEAR A HAT! EVER! He categorically states that he hates them several time throughout several books- I think due to the black hood on his head during the trial re Justin DuMornes death and subsequent doom of damacales as his sentence note he rips the hood of molly during her trial and always says how his head is unprotected unlike what he gets from his duster the hat may be coming as an extra 'mantle' of the winter knight but it's not here yet. In summary. No hat.