Depends on the Demonic Domain.
In some, if you don't dispel them, they get a massive M.Atack boost. In others, it's a massive Attack boost. I think for Kerberos or Aamon, it disabled a special ability of theirs? I don't remember exactly which one it was, I just remember it was one of the ones in the Wrath Fortress...
This can affect the fragments too, not just the Core, which can be relevant in some domains where you might have fights that you can't avoid for whatever reason.
So generally, if you're trying to beat whatever it is without resorting to a collar break, you kind of need to dispel all of those spots, or else the Core will hit like a truck or do something gimmicky that's hard to deal with unless you're somehow perfectly over-specced for that specific core and whatever fuckery it has going for it.