It is not possible for the sun, or rather its mass, to disappear. In
certain circumstances we can imagine it accelerating away rapidly.
For instance, if the sun was large enough to explode as a supernova,
which it isn't, a large proportion of its mass would be rapidly
ejected outwards, after which, in Newton's terms, we would not feel
its gravity. It may be more complicated with Einstein. (If the Sun
is converted to an evenly distributed hollow sphere of expanding gas
containing a vacuum, and the Earth is anywhere in the vacuum space
inside the sphere, Newton's laws say that there is no force of gravity
on the Earth from the mass of gas around it, or, rather, the gravity
of the mass in different directions cancels out. I think the same
applies if it's a ring instead of a sphere, but it must be
symmetrical.)
So, so far, I disagree; the sun's mass cannot disappear from its
present location, all at once. At most, it could move, or explode.
And that demands energy, and that has mass too - sort of.
The sun in fact is very gradually losing mass, both as energy, and as
the "solar wind" of the stuff of the sun blowing away, basically. But
most of its mass will still be there when it expands to red giant
size, then ejects a "planetary nebula", and contracts to a white
dwarf, about five billion years from now.
I'm led to believe that in the meantime it will get gradually hotter,
and in about one billion years all of the water on Earth's surface
will be evaporated, presumably rendering it uninhabitable to life if
that isn't already the case for other reasons.
So much for the Sun, then. And the Earth.
Massive objects moving around in certain ways theoretically generates
gravity waves, but these have not been detected.
Gravity in General Relativity is considered to be not a force but the
shape of spacetime, and objects are not pulled by gravity but follow a
straight line, but a straight line through bent spacetime looks like a
curve.
Some particular implications of the theory of relativity have been
tested by observing odd objects in deep space such as pulsars (rapidly
rotating neutron stars) with companion stars, but I don't know if that
includes the finite speed of gravity.
I gather that a planet was very recently detected orbiting in a binary
star system. Extended study of its motion, which probably would be
extremely difficult, might give a clue as to whether its orbit around
two "moving" stars is consistent with a finite speed of propagation of
gravity.