I'm not thinking that this will happen truly spontaneously, though cascade organ failures would indeed be interesting. More likely the trivial injuries and accidents our dwarves now suffer through will become a billion times worse.
At the moment if your dwarf takes a one z-level plunge in one of those everyday channel digging accidents, he will probably be perfectly alright. At worst he may have a couple light (gray or dingy yellow-brown) wounds that will quickly fade into nothingness and more likely he will be just stunned for a second. With Toady's updates, I expect the dwarf will break his ankle and be stuck in bed or in a deskjob for forever and a day or else get scrapes that get infected and cause your dwarf to turn into a seething mass of puss and skinless muscle that even Papa Nurgle would be proud of. Flakes of obsidian will fly up when you are mining and blind your dwarf in one eye and then, months later, the other. Your dwarf has the nerve damaged on the thumb on his right hand and he can never wield a sword again.
Now this is awesome... or at least it will be once Toady adds in little things like surgery, convalescence wards, civilian protective gear like goggles and thick leather gloves, prosthetic, and benign euthanasia. Until then things might be a little bit of a crapshoot. We will no doubt have whole wards of limbless, eyeless dwarves expelling endless rivers of blood and puss and feces while screaming incoherently for two years until they eventually succumb to their wounds. But that is the price we pay.