Who caused the Burn? A second Terran Empire. One formed sometime after then events of DS9's Mirror episodes.
The basic principle of the Mirror Universe is that relationships and political structures operate as their opposite. Honorable people are dishonorable. The peaceful are warlike. I think it extends to the Federation vs Terran Empire as well. I think as the Federation ascended over the 23rd and 24th century (and beyond) the Terran Empire declined and fell. And hundreds of years as the Federation continued to grow in strength and endure, the Empire stayed dissolved. And then a few hundred years ago the Empire got back together and has been building its strength in the mirror universe. Meanwhile the Federation entered a slow decline as it hit limits. Technological. Political. Even in terms of space itself - Andromeda and Triangulum galaxies are just too far away even for Quantum Slipstream. As Rome hit effective geographic limits of its expansion before its slow decline, so too did the Federation. It harkens back to something T'Kuvma said in the pilot about the Federation's expansionism.
Time travel opened a new domain for Starfleet to explore and the Federation to perhaps grow into. And for about 100 years it looked like it would be the answer. The Temporal Wars put a stop to that. The federation hit a ceiling again.
And then the Burn delivered a big push and shoved the Federation in a slow decline as dilithium dried up, over the edge of a cliff.
So what is the nature of the Burn? I think The Terran Empire reconstituted itself sometime in the 26th or 27th century or so and spent the next few hundred years conquering the entire galaxy. Literally the whole galaxy, not Star Trek's occasional reckless use of the word. An Empire that would make the Empire of Star Wars blush in its scope. But it encountered the same problem as the Federation: what happens to an empire that does not grow? It whithers.
I think the Burn was a misfire of sorts. I think 120 years prior to Discovery, the Terran Empire tried to punch a hole through the increasing distance between the two universes big enough to send an enormous armada through to conquer the Federation's Galaxy (and thus, expand its domain into that).But the attempt at punching that hole through the "great distance" messed with subspace and some of this universe's laws of physics. Every warp core emitting any kind of subspace field saw it's dilithium (its modulator) was destroyed as the effect rippled out from its point of origin (an Episode 6 plot point). Dilithium, already rare, wasn't *actually* at fault per se. The Terran Empire didn't intend to destroy the Federation this way. It was incidental to them trying to open an enormous subspace fissure.
After the failure to punch a hole, the Terran Empire didn't try again, but watch for some relationship with Discovery's wormhole that brought them to the 32nd century having made a second try worthwhile for the Terrans. Which is why Georgiou is having "episodes". That's the Terran Empire trying to bring the universes closer based off something they learned (maybe from the Wormhole... somehow...) having a biological effect on the only Terran in the Prime universe.
Towards the end of the season, we'll get more black boxes and eventually coordinates. Discovery will find the rift and realize what is going on. They'll realize the Terrans are trying to break through. The risk wouldn't be a second Burn pe se (that'll be incidental). It'll be the largest invasion fleet in modern galactic history and versus a Starfleet that has about 15 ships left. And that'll force Georgiou to make choices about her relationships and loyalties. It's also suitably Star Trek-y in that in the far, far distant future, even after a millenium of prosperity, our greatest enemy is still our "original sin" - our dark impulses and baser instincts expressed when humans in the Terran universe killed the Vulcans during First Contact, while humans in the Prime Universe decided on a path better than their baser instincts. And how even though we can never really escape them, we can keep them safely under lock and key by the choices we make (as shown via Georgiou).
But in short... the Burn? It's not the Borg, or bitter old Romulans bent on revenge, or Klingons or Q or any other fanfic. It's something within Star Trek: Discovery's sandbox which relates to the show's leads in its most profound way, and that's the Mirror Universe and the Terran Empire.