Denner resin comes from the sap of denner trees, and addicts ('sweet-eaters') will go to great lengths to eat it. It usually turns their teeth white, as it does for a girl whom Kvothe sees dance naked on the snow as part of a deal she made to get some. It is addictive and triggers euphoria for a time, but it has nasty side effects like numbness, delirium, and disorientation.
There are reasons to doubt that a) Denna symbolises/personifies denner resin, and b) that Denna is a physical manifestation of denner resin. Certainly, Kvothe's narration is imperfect, such as the time he downplayed the incident when he was arrested in book two, but there's a limit to how far Rothfuss could take an extended metaphor before the reader stops taking Kvothe's story seriously.
Kvothe is too damn clever to take a drug like denner resin. Look at what Abenthy, an arcanist, said about him.
"hardly ever makes mistakes. I'll bet he knows every song you've ever sung to him. He knows more about what's in my wagon than I do. [...] It's not just memorisation though. He understands. Half the things I've been meaning to show him he's already figured out for himself. Have you ever known a boy his age who talks the way he does?"
In book two, Elxa Dal is similarly impressed at Kvothe's ability given his age, expressing surprise that he is "already a re'lar" at 17 (although Kvothe is actually 16). This is because Kvothe's knowledge massively surpasses both his age and his time.
Within a span I could identify any chemical in his cart. In two months I could distill liquor until it was too strong to drink, bandage a wound, set a bone, and diagnose hundreds of sicknesses from symptoms. I knew the process for making four different aphrodisiacs, three concoctions for contraception, nine for impotence, and two philtres referred to simply as 'maiden's helper'.
We are told that Abenthy's main interest is chemistry, so Kvothe knows a lot about ailments and remedies, and poisons...
As he [Trapis] spoke he absentmindedly crossed his legs and began to rub one of his bare feet. Inefficient circulation, a long-unused part of me thought. Increased risk of infection and considerable discomfort. Feet and legs should be raised, massaged, and swabbed in a warm infusion of willow bark, camphor, and arrrowfoot.
This was during Kvothe's time in Tarbean, where he spent three years on the streets. Considering he still remembers the things he's learnt, it's likely that he understands the full side-effects of the poisonous denner resin, so it's not plausible that he would become addicted. Kvothe also had an advantageous upbringing. Abenthy describes him as 'living in such an enlightened atmosphere', which is not the sort of place where you find drug addicts. Look at the unfavourable way in which Kvothe describes the so-called 'sweet-eaters' (as opposed to the pleasant way he describes Denna, see below):
I learned to run from anyone with an unnaturally white smile. [...] Tarbean is full of dangerous people, but none as dangerous as a sweet-eater filled with a desperate craving for more resin. They will kill you for a pair of pennies.
In addition, Kvothe only just manages to scrape a living in Tarbean. He could not have been such a good pickpocket if he had been eating expensive drugs, and considering how impressed he is with a low-value coin, he needed all the money he could get.
I gaped. A silver penny was worth ten copper pennies, or fifty iron ones. More than that, it was worth a full belly every night for a month. [...]
Through dangerous trial and error I discovered the proper way to slit a purse and pick a pocket. I was especially good at the latter. Locks and latches of all kinds soon gave up their secrets to me.
It's not only that obtaining denner resin would have made him poor in the first place, but the side-effects of eating it would hinder his ability to escape the brutal guards inside Tarbean. As it is, Kvothe survives against the odds because of his unnatural wit and sensitivity, which he would not have if he had been insensible as a result of taking drugs:
Someone was watching me. On the street you either develop a sensitivity to certain things, or your life is miserable and short.
Arguably, Kvothe's intelligence is offset by the sheer stress he suffers throughout the trilogy. His family was murdered, he's in debt with a gaelet, and a really rich guy hates him. But think of it this way: would Kvothe be more likely to be daunted / putt off denner resin if he knew all about it, or if he had no idea what it could do?
Kvothe is too good at the lute to be a drug addict.
I could play for hours upon hours [...] I began to play something other than songs. When the sun warms the grass and the breeze cools you, it feels a certain way. I would play until I got the feeling right. I would play until it sounded like Warm Grass and Cool Breeze.
Kvothe also wins his silver talents at the Eolian for his grand performance of 'The Lay of Sir Savian Trailard'.
Denna changes her name habitually:
Dianne, Denna, Dinnah, Dyanae, Dinael, Dinay, Dianah, Donna, Dyane and Alora.
Just because Kvothe first knows her as 'Denna', it doesn't mean that that it important enough to be taken as a literal play-on-words for denner resin. She herself dismisses the name 'Denna':
'Denna', she said softly. 'I'd almost forgotten her. She was a silly girl.'
Denna affects her surroundings too much to be a figment of Kvothe's tweaked storytelling.
Roent's wife, Reta, sat in front of that wagon. Her mien wavered from severe, when she watched the men loading the wagons, to smiling when she spoke to the girl [Denna] standing nearby.
[...] Bast held up his hands defensively. 'It's just something I noticed, Reshi. All the women in your story are beautiful. I can't gainsay you as a whole, as I've never seen any of them. But this one [Denna] I did see. Her nose was a little crooked.'
[...] 'Here we all are.' Sovoy smiled at me as he walked over and put his arm casually around Denna's waist. [...] Together, they turned and walked arm in arm back to their table.
[...] Three more days and five more fruitless trips to Imre. Neither Deoch nor Threpe had heard any news of her. Deoch told me that it was her nature to disappear like this, and that looking for her would serve about as much purpose as calling for a cat. I knew it to be good advice, and I ignored it.
[...] We were almost out when the man behind the bar called out. [...] 'That your cousin [Denna] then?' he asked. 'Has the constable said she could go?
[...] Denna: 'an older gentleman introduced himself to me. [...] If things continue smoothly, I think he'll be my patron before the year is out.'
[...] I was about to explain things to her when Denna gestured to a man standing behind her. [...] 'Kvothe, this is Lenthoren.'
[...] Simmon: 'I talked with Sovoy. He's still not over her [Denna]. He really loved her, you know.'
If Denna isn't a real person, then the people apart from Kvothe whom she affects are either made up, which would be extreme and contrived, or many scenarios are conjured up merely to accommodate her, which would undermine the point of Kvothe's narration. However, this particular factor only discounts the idea that Denna is fake, not the idea that Denna is as bad as denner resin.
Denna is characterised to fully to be fake/ a negative force like denner resin, worthy of being ignored.
'I had pneumonia when I was just a tiny baby [...] That's why my lungs aren't good. It's horrible not being able to breathe sometimes.'
If Denna was a made-up character to hint at Kvothe having a drug addiction, or she was a negative force like denner resin, it's not plausible that she would have been given a backstory that invokes the reader's sympathy. Consider her following descriptions. They sound sincere, as if Kvothe really is romantically interested in Denna, and she likes him back at least a little bit. It would be misleading if Rothfuss had intended Denna to be as detrimental a force as denner resin.
She smiled at me then. It was warm and sweet and shy, like a flower unfurling.
[...] This time as I sang it I looked out into the audience, hoping at the end I would hear a voice answering my own. [...] Then a voice drifted onto the stage, gentle as a brushing feather, singing [...] She sang as Aloine; I as Savien. Part of me wanted to search the audience for her, to find the face of the woman I was singing with. [...] Distracted, I touched a wrong note and there was a burr in the music.
[...] Then I heard a voice, a voice like burning silver, like a kiss against my ears. Looking up, my heart lifted and I knew it was my Aloine. Looking up, I saw her and all I could think was, beautiful. Beautiful.
[...] Denna: 'You remind me of a Willow', she said easily. 'Strong, deep-rooted, and hidden. You move easily when the storm comes, but never farther than you wish.'
[...] I sat there in the dark, holding her sleeping body in my arms. She was soft and warm, indispensably precious. I had never held a woman before.'
[...] But tonight she was wearing a dark green dress that left her shoulders bare. She was stunning. She knew it. She smiled.
They even speculate about her patron, who is confirmed by the Cthaeh in book two to be beating her. Firstly, the fact that the all-knowing Cthaeh acknowledges Denna's existence discounts one of the theories that Denna is made up. Secondly, if she is made up/ just a symbol of denner resin, then her patron would have to be made up too. It's unlikely that Rothfuss would have created a whole character to accommodate a false character/ a purely symbolic character.
Now that I looked more closely, I noticed that she had a bruise on one temple in addition to the bandage on her arm. [See above for the mention of her patron]
Denna is treated like a real human. She eats sociably and even saves Kvothe's life, despite that these things undermine the idea that she is a negative drug/ a false character, making it implausible that Rothfuss should have put in the following scenes:
Denna and I had the opportunity to eat some lunch of our own. Just some flatbread, sausage, and the rest of my carrots.
[...] My hands gripped at the top of the arch, but couldn't find any purchase. Denna caught me.
Denna is also observant in such a way that a negative / fake character would have no need to be, pointing things out to the protagonist (sometimes before he notices them) in a way is either affectionate or helpful:
'Did you know that when you're angry your eyes [...] I thought I was imagining it before [...] but your eyes really do change colour.'
Kvothe says that she is the only one to have noticed this, apart from Abenthy who is sharp-witted by virtue of being an arcanist.
[Straight after Kvothe nearly dies by losing his grip on a cliff] 'Oh!', she gasped, her hands going to her mouth, 'Your beautiful hands!'
Of course, the same could be said even if Denna was a negative influence on Kvothe, but I think she shows too much forthright compassion to be seen as such. It is unlikely that she is a symbol of addiction for the same reason (because a drug addiction wouldn't save his life like Denna).
Denna is too smart (in a positive way) to be a symbol for denner resin / a negative force.
Denna has a high level of intuitive intelligence, allowing her to suggest a plan when they're in danger: lure the draccus off a cliff.
'Do you think that would hurt it?' I asked dubiously. 'Well', Denna said, 'when you flick an ant off the table it doesn't get hurt even though for an ant that has to be like dropping off a cliff. But if one of us jumped off a roof, we'd get hurt because we're heavier. It makes sense that bigger things fall even harder.' She gave a pointed look at the draccus. 'You don't get much bigger than that.'
She was right, of course. She was talking about the square-cube ratio, though she didn't know what to call it.
Denna eats denner resin by mistake, and that would either make no sense or be very contrived if the theory the she symbolises denner resin were true. However, this has nothing to do with the theory that Denna is a negative force.
Kvothe is concerned about the resin she has eaten, implying that he would have no intention of consuming it himself.
She had pried a large disk of sticky material out of the bottom of one of the pans and taken a bite out of it. [...] 'What's ophalum?' she asked softly. 'A drug. Those are denner trees. You just had a whole mouthful of denner resin.'
[...] Denna: 'I suddenly feel really good.' [...] 'It's the resin', I said.
[...] 'I want to get you back to Trebon', I said. 'Just in case you ate more resin than is good for you. I wouldn't trust any doctor living there, but they probably have some medicines I could use. Just in case.'
[...] I was no longer worried about her overdosing. After gathering a small mountain of firewood, her mania was fading, leaving in her in a content, almost dreamy legarthy. Still, I knew the after-effects of the drug would leave her exhausted and weak. I wanted her safely in bead in Trebon for her recovery.
Kvothe is careful when approaching her, and when speaking of her. She seems far more sophisticated / complex than a drug. Everyone knows that drugs feel good at the time, but has bad side-effects, end of story. Denna is far more baffling.
As with all truly wild things, care is necessary in approaching them. [...] With slow care rather than stealth we must approach the subject of a certain woman [...] I fear approaching her too quickly even in a story. Should I move recklessly, I startle the very idea of her into sudden flight.
Kvothe gets angry when Chronicler records his attempts to summarise Denna, despite not indicating otherwise. Drugs is not a topic you would skirt around in such a squeamish, touchy fashion. Drugs are drugs. A victim of drugs would not want to drag it out. The sooner they are covered by the narrator, the better (from the narrator's perspective).
'Her lips were always red, morning and night. As if minutes before you saw her, she had been eating sweet berries, or drinking heart's blood. [...] What good can come of this? How can I make any sense of her for you when I have never understood the least part of her myself?' / Chronicler had written most of this out before he realised that Kvothe had probably not intended him to. [...] Kvothe's eyes caught and held him. They were the same dark eyes that cronicler had seen before. Eyes like an angry God's. [...] Kvothe tore the half-written sheet with slow care, the sound bleeding he colour from Chronicler's face.
Kvothe treads carefully with Denna, as if she is delicate (unlike a euphemism for drugs), and as if her nature is unpredictable, unlike drugs, whose effects are continuous and predictable:
I knew the truth: I would never see her again. But here she was, and I was entirely unprepared.
What next> A kiss on the hand was traditional, but what sort of kiss was appropriate? In Atur you merely nod over the hand. Cealdish ladies like the moneylender's daughter I had chatted with earlier expected you to brush the knuckles lightly and make a kissing sound. In Modeg [...]... I pressed my lips gently to the back of her hand for the space of time it takes to draw a quick breath. Her skin was warm and smelled vaguely of heather.
[...] We talked through the long hours of night. [...] It was like we were doing one of those elaborate Modegan court dances, where the partners stand inches apart, but - if they are skilled - never touch.
[...] I should have been bolder and kissed her at the end. I should have been more cautious. I had talked to much. I had said too little.
This point also does not discount the idea that Denna is a negative force.
In conclusion, I think that all the Denna/denner-resin theories are unlikely. Congratulations on reaching the end of this answer!