Although, to be honest, given the size of the death star relative to the size of a planet, I'm not sure that long term instability is something I understand as physically possible. Let's say 1 million cubic kilometers have to be removed, and you need to dig out several times that amount to get at it. Seems like a lot until you realize an earth like planet like Ghorman has 260 billion cubic kilometers of material. Biosphere collapse, sure. Maybe something like the multiple Deccan Traps events releasing massive amounts of toxic chemicals and ash into the atmosphere. But, the Death Star just doesn't have enough mass or size to significantly alter the mass of a planet.