Documents show the single sculls gold medallist was allowed to take suppositories to help treat hemorrhoids.
Listen: Mahe Drysdale on the Crowd Goes Wild Breakfast
Burling, who won gold with Blair Tuke in the men's 49 in Rio, had a TUE for the prohibited substance remifentanil for a wisdom tooth extraction in September last year. Burling said he's not even sure if he used the drug.
Drysdale told the Crowd Goes Wild Breakfast on Radio Sport if athletes apply for a TUE, then it should be public.
"I'm disappointed that our confidentiality has been hacked, it's illegal what they've done. But on the other hand I think it's a good conversation starter because my feelings on it are if you apply for a TUE, then it should be public. Have some transparency there. My personal opinion is if you're going to use a substance that others can't use you should be able to justify and be happy for it to be out in the public domain. That should actually help the TUE system not get abused," Drysdale said.
"I have no issue and I feel comfortable standing by and explaining why I had those TUEs and I think every athlete should be."
Drysdale said the last time he used the drug that needed a TUE was at last year's Rowing World Championships and the only reason he had exemptions granted this year was in case the issue flared up again and he could use the treatment straight away instead of waiting three days for an TUE clearance.
"I've suffered from hemorrhoids the last three years and I've tried different treatment methods and nothing has helped too much. Usually I only suffer around racing time and this year I had two TUEs, earlier in the year and again around the Olympics only on the off chance I had hemorrhoids."
Burling told Newstalk ZB it's a very lengthy process to make sure it's not a performance enhancing drug.
"It's an interesting headline to wake up to," Burling told Newstalk ZB.
"Haven't really been woken up to being outed by a Russian hacker before, but honestly it doesn't worry me at all. I've got nothing to hide. Yeah I've had a TUE, a wee while back when I got my wisdom teeth out and they're the only thing I can think of that would be anything at all interesting."
Listen: Peter Burling on Newstalk ZB
Burling said he first became aware of his name being leaked by Fancy Bears was when Newstalk ZB contacted him.
"It never really occurred to me that I would be named in one of these kind of things. I'm really fortunate that I'm in a sport where it's not a very common thing, drugs. I don't think anyone has ever been banned from sailing for that kind of thing."
The leak also includes data from double Olympic triathlon champion Alistair Brownlee.
Brownlee was granted an exemption to take Diamox to treat altitude sickness after climbing Mount Kilamanjaro in 2013.
Previous athletes to have their TUE data leaked, after the WADA (World Anti Doping Agency) computer system was hacked, include Olympic champions Serena Williams, Simone Biles, Mo Farah, Rafael Nadal and Justin Rose.