I think “burning” is a metaphor that has actually been employed by both sides.
BoS, in FB hack – “You can’t silence us, we will release one name a day, until this entire organization BURNS to the ground.”
Call to JoeKaye, presumably from BoS – “Everything will burn. Get out while you can. It’s all over for you and them. Don’t go back to the site. It will all burn.”
Overseer, describing his job watching over (presumably) GK4 – “I am here to make sure She doesn’t burn down the house.”
References to Rizzo, from both Overseer and Gatekeeper 4:
“he will be given a second chance but first he must find a key to re-enter that which he has burned.”
“You’re burning, aflame in another solar system wondering why this keeps happening to YOU. Alright, I’ll admit it… your natural destructive chaos does have a certain… sumpin’ sumpin’.”
I’m inclined to think the “burning down the house” metaphor is just a favored turn of phrase among OOA and BoS leadership.