This story had a modern (late 90s, early 2000s) setting. A boy worked at a zoo/sanctuary for dragons in a world where dragons were a real, exotic animal and not a fantastical creature. He ventures into the woods and finds a dead mother dragon and an infant dragon which imprints on him. He hides this dragon in his shirt to keep it warm, as it grows it becomes so hot it scalds his stomach and gives him burns, which when discovered by a doctor leads him to be given eczema medication. Later in the story he enters a gigantic dragon cave with a huge black dragon. If I recall correctly there was an element of telepathic communication between these dragons and the boy.
This is Dragonhaven (2007) by Robin McKinley.
From a review:
Jake is the first-person narrator and the center of this story. He lives with his father and a small group of park rangers and other scientists at Smokehill National Park, one of the few remaining reserves for dragons in the world. McKinley's world is much like ours, with the same politics and governments, except that dragons exist: huge, flying dragons that breathe fire. But they've been hunted to near-extinction, including via an actual war in Australia, and are now dwindling in a small handful of isolated reserves. No one is entirely sure how many are left since, for huge creatures, they're remarkably hard to find and document. The Smokehill staff mostly estimate them from prey kills and other indirect signs.
The story proper starts when Jake, on his first overnight trip in the park, encounters a dying dragon and a dead poacher. This is a catastrophe for the park: the dead poacher will provide ammunition for everyone who thinks dragons are too dangerous and should be wiped out. But for Jake, it's a more immediately life-changing event. The dragon was a mother with newborn infants, one of whom has survived and who promptly bonds with Jake.
This may sound like it's moving back into traditional fantasy territory, but it's not. Jake's relationship with Lois (the name he gives the baby dragon) is very unlike fantasy bonds with baby hatchlings and quite a bit like taking care of an actual infant. Except the infant is hot enough to burn the skin. And wants to be in a pouch (McKinley's dragons are marsupial, sort of) and therefore needs to crawl inside Jake's clothing and press against skin. And that's not mentioning the smell. Or the excretions.
The infant 'dragonlet' is already hatched from its egg when the protagonist, Jake, finds it, but this book appears to match your description in every other respect, including the eczema...
After that it was always the same doctor, and after a while he wanted to write some kind of paper on my skin complaint, which he wasn't even sure was eczema, he said (bright of him), and he sure tried to get me to come up to some hospital and have some fancy tests done, but I didn't want to go (leave Lois overnight?) and Dad wouldn't make me, obviously, and since I was healthy except for the eczema, the doc reluctantly let it go.
... the cavern full of dragons...
Yes. You knew this already, reading it here, but I was having a lot. of trouble with reality. We were in a cavern full of dragons.
I'll let that sink in for a minute. It takes a lot of sinking in. Think yourself out of your comfy chair and your nice house with the roads and the streetlights outside - and the ceiling overhead low enough that a fifty-foot dragon can't stand on her hind legs and not bump her head - and think yourself into a cavern full of dragons. Go on. Try.
... the large, black dragon...
It was hard to tell dragons from rocks and shadows, and while I was never sure about this either it seemed to me that it wasn't always the same dozen or fifteen dragons - although I thought Gulp was nearly as big as most of them. The one I could see most clearly, however, was a lot bigger than Gulp, facing us from the other side of the hearth. He was black, with no iridescence at all, although on some of him - eye ridges and nose, spine, elbows - the scales were outlined in red. I had thought Gulp was scary - he was scary. He made Gulp, look like a cuddly toy dragon. A fifty-foot cuddly toy dragon.
... and the dragons being telepathic.
I can't tell you how bad this was, how lost I was, how mindsmithereeningly alone I was, in this flickery shadowy red-purply nowhere, full of huge breathing shadowy things with huge shining eyes. And there I was, scared silly, scared beyond silly. And one of the things I came out of that experience with is a total inability to use the word "telepathy. " It just doesn't fit, okay? And also . . . telepathic dragons. Pleeeease. That is so last century. I've got like shelves of Mom's old story books with telepathic (if pouchless) dragons in them.