I've read this one :) It's called A Belly Full of Fire by FruitPastilles.
I was able to find it because this was such a unique fanfic that I remembered part of the summary, "he breathed fire". Here's the summary:
When people asked Hisashi what his Quirk was, he'd flash a grin and say he breathed fire. It was a small lie, easier to say than 'my human skin is too small and some bits slip out' despite the fact he never checked whether or not he had that toe joint. Izuku wasn't born Quirkless. He was born something else.
And here is the part you remember:
Izuku ran forward, desperation fueling his sprint, his chest burning with exertion as he crashed to his knees in front of them. The robot had one lumbering step left.
Close up the rubble was bigger, seemed more than he could handle.
"Here!" Izuku gasped, grasping at the girl's wrist. "I'll help you!"
The burning intensified.
The sun dimmed and disappeared as shade was thrown over them, their light blocked by that one, lumbering footstep.
"Oh," the girl whispered, voice warbling sadly, fearfully. "We're going to – "
"No!" Izuku replied, standing up and letting her go, fist clenching by his side. "We're not!"
The burning was white hot in his chest, like One For All, his new Quirk, was telling him it was ready.
It was not One For All telling him it was ready. It was something else.
With a ferocious bang, like a bomb had gone off, the entire arena was engulfed in smoke. Through the smoke, something whistled as it shot by, and a piece of the Zero-Point robot's head embedded itself into a nearby building.
In the examiner's officer, several bodies leaned forward, interested.
Nobody could be prepared for what the dissipating smoke showed.
Where there had once been one (1) Giant Robot and two (2) endangered examinees, there was now a very large, very scaly dragon.