This question has been asked many times in conjunction with a suddenly disappearing sun. The common reaction to this is that it takes a while for the spacetime curvature to be changed around it. In accordance with special relativity, so it is said. An instantaneous change from curved to flat by means of a disappearing mass doesn't mean though that information is traveling faster than light.
Why not? Simply because you can't remove a mass instantaneously. This renders the question meaningless maybe, but you can do it in the imagination. If the mass suddenly is not there anymore, there will not emerge a gravitational wave. Only when you move it this will happen. Which is needed to transmit information. If I move the mass fast to and fro, a wave will emerge, so you can convey information with it. But not if the mass suddenly ceases to exist.
In connection with charge and the electromagnetic field surrounding this charge, I've read about experiments to uncover if the change in the field surrounding the charge is instantaneous. This means that there is a doubt for that case. The same arguments I gave above can be applied to the electromagnetic field. Will an electromagnetic field instantaneously appear if charges appear in the creation of a positron-electron pair (or disappear when they annihilate)?
See also this article (written in 2006 by W. Engelhardt, retired from Max-Planck-Institut für Plasmaphysik, D-85741 Garching, Germany), in which it is written:
the instantaneous transmission of both energy and information over macroscopic distances is feasible by using the quasi-static fields which are predicted by Maxwell’s equations.
I don't think that the conclusion drawn there (or by me) is correct though (information can't travel at the speed of light). Even if you could create single electrons (contrary to an electron and a positron, appearing at a point), and the field of that electron would be present at the same time everywhere, you could use them to transmit information: when I create one electron per second it's an A, when I create two per second, it's a B, three per second a C, etc. but is Engelhardt mistaken too? Is it because he's retired..., maybe?