The biggest thing would be signs of burn marks on fossil bones and the inner side of dragon teeth
The only parts that reliably preserve in fossils are hard tissues. Bones, teeth, etc. In rare cases you get mummified skin or soft tissues preserved, but this is rare. As a result, anything that isn't preserved in bones is very difficult to figure out.
This goes for most of the squishy stuff that people have recovered from fossils. Muscle size and usage in feeding and locomotion comes from attachment scars on bones. Evidence for diet and migrational patterns comes from differences in chemical isotopes in bones. Behavioral inferences like herding or parental care come from unusual associations of bones. Dietary habits come from comparative anatomy or bones/shells within a rib cage. And so on.
The idea that dragons breathed fire would best be inferred based on something left on bones. Burning leaves marks on bones that are inferrable even after fossilization. In paleoanthropology, such evidence is used to infer human usage of fire. Evidence of burning in weird environments might imply fire-breath, though that requires accurate inference of paleoenvironments where forest fires are rare (e.g., a rainforest or a desert). Even then, researchers might interpret dragon burns as evidence of infrequent forest fires. A skeleton where only part of the body was burned as if hit by a flamethrower rather than all over as if exposed to a forest fire would be hard to explain. Additionally, these burn marks would only be on prey or other animals, you would rarely find burned wood or charcoal like you would expect if the burns were from forest fires.
The biggest sign that dragons breathed fire would be their teeth. Because dragons breathe fire from their mouth, the inner face of the teeth would likely be singed because that's where the fire is coming from. I believe Dragons: A Fantasy Made Real and Dragons and Unicorns: A Natural History pointed this out. As reptiles, dragons would likely replace their teeth regularly (especially if they were regularly being exposed to flame), and would be pretty common in the fossil record. These shed teeth would show signs of being heated on one side, signs of charring or being stained by smoke, or unusual heat-resistant enamel.
This would be true even if dragons aren't really breathing fire and are really breathing some bombadier beetle-like chemical or vomiting stomach acid. The teeth of humans with bulimia or who frequently engage in vomiting show signs of acid wear on the inside face of their teeth due to exposing them to fairly acidic vomit. Dragon teeth would look the same.
Eventually, the combination of signs of fire being used in a manner more similar to flamethrowers than forest fires in paleoenvironments that really have no business having forest fires and burn marks on the inside of dragon mouths would suggest to researchers that dragons are the cause, and that somehow the fire is getting inside the dragon's mouth. Causation or correlation wouldn't be clear at first. Some kind of fire-breathing mechanism would be considered more likely than dragons being so stupid they tried to eat fire.
Isotopes might also help if dragons have some kind of weird metabolism in order to breathe fire, and store unusual carbon/oxygen/nitrogen isotopes or elements in their bones. However, isotopic studies have really only come into vogue in the last 15 years in paleontology, so depending on the techbase of your setting it might not be reasonable.