Imagine you could delete the sun from its existence at an instance. Information cannot travel faster than the speed of light so the earth will keep orbiting the sun for about 8 minutes without knowing the sun is already gone (the distance between the earth and sun is ~ 8 lightminutes). After these 8 minutes the information that the sun is missing will reach the earth and the earth will fly away on a straight path.
So what will happen if we did this to a black hole? The curvature near a black hole is so extreme that information cannot travel outwards inside the event horizon. So if we removed the mass of the black hole in the center does the outside world know that we removed the mass?
And a related question: If we have a black hole with charge $+Q$ and we add to it a charge $-Q$, does the outside world know that the black hole is now uncharged? Again this addition is done instantaneously, without this charge ever crossing the event horizon. The difference with the previous case is that the changing the charge doesn't cause the curvature to disappear so this might result in a different outcome.
To address the issue concerning 'the impossibility' of this setup: this process is impossible in real life, but at least for the case of the earth-sun system there should be no issues mathematically. You have some matter distribution $\rho(x^\mu)$ which, by the Einstein field equations, determine how the curvature of the spacetime surrounding it evolves. Setting $\rho$ to zero still allows a solution albeit with discontinuities. Which you can circumvent too.