The situation is too ill-defined for an answer.
The problem there is that in general relativity, you do have general conservation laws that follow from the Einstein field equation. In the asymptotically flat case, you have conservation of a global ADM mass, and in all cases there is a local covariant conservation law that requires the stress-energy to be divergenceless. Stess-energy must move locally; it can't just disappear at one place and move to another.
That means that if the very massive object simply winks out of existence or is teleported to far reaches of the universe, then we can't use general relativity to predict what happens, because such an event blatantly violates it already. Thus:
Or does the magical disappearance of the massive object have immediately observable effects on the small object?
Newtonian gravity doesn't care about conservation of mass, but general relativity does, or at least has roughly analogous laws that prohibit this type of situation. If one starts invoking magical disappearances or teleportations, then there is no reason for the same magical event to not make the spacetime instantly flat either. It doesn't make any sense to apply a theory to a situation that directly contradicts it, so the question of what happens becomes ill-defined.
There is a sense in which you can make the matter of the very massive object disappear by collapsing it into a black hole, since (isolated) black holes are technically vacuum solutions of GTR, and it gets around limitation of a divergenceless stress-energy because singularities are not part of spacetime. But in that case the other body could still orbit it in just the same way.
Perhaps one could drop the matter into a wormhole. I don't know how that would work out in general, but at least for the flat "Lorentzian wormholes" investigated in Visser's monograph, the mass you transport in this manner gets added to the wormhole mouth. So you'll have a very heavy wormhole end, which the other body would still accelerate toward it.