For calculation purposes, let's say it's a cone with cone angle (at Earth's center) of 30 degrees, that extends down to the outer surface of Earth's solid inner core, and the entire mass is ejected over the span of one minute. I have some educated predictions, but I would really appreciate some scientific, source-backed corrections of any faulty assumptions/conclusions!
Also, feel free to redirect me to a different forum if this Exchange isn't really the place for sci-fi "what if" questions like this.
My predictions
Fate of the ejected material
Using Earth's equatorial radius of 6378 km, then the volume of the cone (technically a spherical sector) down to Earth's center is $$ \frac{2\pi}{3}r^3(1-\cos\phi) = \frac{2\pi}{3}\times (6,378,000 \textrm{ m})^3 \times (1-\cos(15^\circ)) \approx 18.5 \textrm{ billion km}^3; $$ the actual "cone" (truncated spherical sector) from my original description would be a little less than this when we subtract the volume inside the inner core. Given that Mt. Everest has a volume of ~90 km^3, we're talking about a blob of material about the size of ~200 million Mt. Everests!
The material will experience friction with the "walls" of the cone and within itself as it ejects, so the initial cone will break up into smaller "chunks" moving at different speeds. Earth's liquid outer core is about 5150 km deep at its inner boundary. If material gets from there to the surface in one minute, then that's an acceleration of $$ \frac{2d}{t^2} = \frac{2\times 5,150,000 \textrm{ m}}{(60\textrm{ s})^2} \approx 2.86 \textrm{ km s}^{-2}, $$ and the material will be moving away from Earth's center at $$ \sqrt{2ad} = \sqrt{2\times 2,860 \textrm{ m s}^{-2} \times 5,150,000 \textrm{ m}} \approx 172 \textrm{ km s}^{-1} $$ by the time it reaches the surface. This is faster than Earth's escape velocity (~11 km/s) but slower than the Sun's (~620 km/s). Some chunks near the boundary will rain back down to the surface as massive boulders and "splashes" of lava, creating massive impact craters and lava fields (depending on the density of the material and max height reached); others will go into a highly elliptical geocentric orbit, many of which will collide with Earth as meteorites when they return; others will go into a highly elliptical heliocentric orbit with apoapsis at the Earth's orbit, and Earth will experience meteor showers (and meteorites of various sizes) as it intersects that orbit again every year.
Fate of the hole left behind
Around the ejection site, a circular region of Earth's crust will buckle outward, like the shell of an M&M being popped out from the inside. Sections of this crust will crack and split apart; some sections will fall all the way back onto the surrounding crust, crushing any landforms underneath, while others will fall forward again and tear off into the void left by the ejecta. The hole itself will begin to refill almost immediately from the bottom up. First, the liquid outer core will flow into the void (orange material in my drawing above). Then, viscous chunks of the inner mantle (dark red) will break off from the walls and float down the flowing liquid core, followed by more solid chunks of rock and sediment from the outer mantle and crust. As all this material flows into the hole, the surface of Earth overhead will sag inward, forming massive cracks and faults. It goes without saying that the entire surface of the earth would be rocked by unimaginable earthquakes and tsunamis.
In the end, the void would be replaced by a deep funnel-like valley ~2500 km deep (a little shallower than than the current surface of the outer core), and maybe twice as wide as the original ejected cone. At its base would be large lakes of lava amid hills of igneous rock. The funnel's wall would be glowing red hot near the base, but less so the higher you went. It would also vary in steepness over its height: lowest near the base where the liquid core was more fluid, then steeper (perhaps as steep as sand dunes) in the middle where rocky mantle material slid down, to less steep near the surface where it was more crust sagging than material flowing or breaking off. The crust around the rim of this valley will be highly irregular, with jagged crags and oddly shaped mountains where the crust broke and slid toward the funnel during its formation. I have no idea how long it would take this funnel to form or cool however, as I can't find good estimates of the viscosity/flow rate of the mantle and inner core. I imagine it would take at least centuries though, if not millenia.
Atmospheric effects
Immediately after the ejection, air would "drain" into the void left behind, creating hurricane-force winds at the surface around the hole. Much of the air above of the cone would've been blasted out into space with the ejected material. Massive clouds of dust and debris would be kicked up by the initial ejection; their spread would initially be reduced by suction of the air flowing into the hole, but eventually the turbulent atmosphere would settle, and global circulation would move the dust around the world, likely reducing sunlight and kicking off a global winter. The atmosphere would also end up lower in pressure (indicated by the lighter blue color in my drawing above), now that its filling a larger volume. The atmosphere currently has a volume of ~10 billion km^3, so if air completely filled the hole, then the atmosphere's volume would have almost tripled (according to my rough calculations above). More likely, it would be ~1-10% less dense, as air would not completely fill the hole, but remain in a relatively thin layer "coating" the hole's walls like the atmosphere coats the Earth's surface today; i.e., the atmosphere would end up with a "valley" of its own.
Weather patterns around the funnel valley (once it's formed) would be fairly normal, with wind circulating in and out of the valley. Like the deep ocean, there would probably be a depth in the valley below which air is relatively stagnant and mixes less with air and water vapor from around the world. Air deep within the valley would also get heated by all the lava and mantle rock, so there would probably still be strong winds down there due to tight convection cycles of air.
If we assume the initial cone was centered on the equator, then its ~30° angle (1/12th of a circle), would mean that the deepest parts of the valley only get ~2 hours of sunlight. This low light, combined with the poor atmospheric mixing, and extremely hot and unstable ground at the valley base, would make conditions down there very difficult for life. Ignoring Earth's curvature, dust and water vapor in the air currently prevent seeing beyond ~20 miles with the naked eye. Even with the air thinner after this ejection, you would not be able to see the other side of the funnel valley if you were standing on the rim; it would feel more like standing on the side of an endless hill looking up at the sky (though I'm not positive, cause you're only looking through the air on "your side of the valley", beyond that is empty space within the void...so maybe you could see the other side?). With the sun directly overhead, that sky would be the normal blue, but darker as you looked down. With the sun at an angle, you would see the shadow of the valley rim extending through the air in a Tyndall effect. You would not be able to see the glowing lava at the base of the valley, even at night, due to air.
Gravitational effects
With the huge amount of mass ejected in the cone, Earth's center of gravity would no longer be at the geometric center of the spheroid, but skewed towards the "full" part of the planet. Thus, you would feel slightly heavier on the opposite side of the planet from the hole. If you picture the gravitational field lines (in my drawing above, black for the old center and white for the new center), then as you walk from the opposite side of the planet towards the valley, the "gravitational tangent plane" of the surface would be under the physical tangent plane of the surface, so it would feel like the ground is getting slightly steeper. This might be balanced near the valley by the physical surface sloping down also. Traveling into the valley, the steepest parts of the valley wall would still feel like almost vertical drops; however, gravity would get weaker as you approach the core, and walking on the base of the valley (if you could survive the heat) would probably feel like bouncing on the moon.
