This turned out unexpectedly fun... and for once the answer to a question involving 99.99% of the speed of light isn't "everybody dies"
Where is this sphere being fired from?
If it's from outside the solar system then hitting the earth as a sphere is.... a problem...
The volume of 1000 kg of tungsten = 51.9 L
Radius = 0.23 m
That's 5439.5 moles of tungsten.
As the sphere approaches from the orbit of Pluto we can estimate how much matter it would hit.
We can treat the space it passes through as a cylinder with radius 0.23 m and height of 7.5 billion kilometers.
It would take our projectile about 7 hours to travel that distance.
It would pass through 1246 km^3 (cubic kilometers) of space.
In the solar system with the solar wind the density of atoms is 2x10^7 per cubic meter, mostly hydrogen or helium.
Treating it all as hydrogen for simplicity that gives us 0.04171 mg of hydrogen.
This gives us 261.3 GJ (gigajoules) or 72.59 MW h (megawatt hours) as the approximate energy involved in the collisions between the fine mist of gas in the solar system and the bullet.
That's the energy of the atoms hitting the front of the bullet, and most of the energy would be effectively dumped into the metal.
Given this is over 7 hours that means there's something like 10.37 MW of energy being pumped into the sphere every hour.
Tungsten has a heat of vaporization of 800 kJ/mol, so it takes 4,351,600 kJ of energy to turn 1 ton of tungsten into gas.
Unfortunately your tungsten "bullet" sphere is getting hit with 37,332,000 kJ of energy per hour so within the first 7 minutes of its 7 hour journey it's become a cloud of atoms glowing hot at about 5555 degrees Celsius...
At this point the calculations get harder because it's no longer a nice neat sphere; it's a cloud of super-high temperature gas traveling so fast that it's glowing like the heart of a star and it's very hot gas so the cloud is expanding very fast. It's now hitting even more of the random atoms in space as it's approaching earth. Assuming the sphere was aimed to hit earth perfectly dead center I can't even tell you if all of the gas would actually hit the earth or if the hot cloud of gas would expand from itself fast enough to mostly miss the earth. I don't know how much extra energy the cloud would lose to hitting atoms along the way after the sphere melts and turns into gas since it now has a massive surface area...
But let's say that the cloud all still hits the earth's atmosphere, but over the 6 hours and 50 minutes since it passed Pluto it's spread out to hit the entire facing side of the earth fairly evenly and none misses and it's still carrying most of its energy.
As in L. Dutche's answer, according to Wolfram Alpha, the relativistic kinetic energy of such a bullet would be 6.265 x 10^21 J, or 1.5 million megatons.
... but... The total power output of the Sun hitting earth is about 4.3 × 10^20 J per hour hitting atmosphere on the side facing the sun.
The bullet is carrying far less energy than a single days worth of sunlight.
Cosmic rays can't pierce the earths atmosphere and the bullet is now more similar to a giant cloud of cosmic rays, but each with far less energy than the "Oh My God particle".
Everything in orbit on that side of the earth would be hit with a huge dose of cosmic rays.
The hard radiation would be caught by the atmosphere and some heat would make it to the surface... but the atmosphere is vast and that much isn't even enough to raise the temperature of the gas on that side of the planet by 1 degree Celsius average.
I suspect anyone outside might get flash burns. I don't know if the energy would be enough to start major fires. To an extent the more energy ends up as high energy particles and radiation the less hits the earth surface as heat.
But to answer your question: As it approached the gas cloud that used to be the bullet would burn brightly in the sky as it impacted gas and dust ... then for a brief moment the entire sky of half the earth would blaze with light thousands of times brighter than sun... it might burn things on the surface.... it might shower a significant amount of secondary radiation... but the tectonic plates would be safe.
Edit: re Yakk's comments below, as the cloud is passing through space there may also be bursts of something like space-lightning as the cloud interacts with subatomic particles that strip away electrons and depending on how far away from earth the bullet starts ..
Napkin math says this means a spread of E-5 (i.e., every E5 meters it spreads out 1 meter), which means over 100 AU that is 150,000 km. That is order-of-order-of-magnitude size of Earth.
So the cloud may partly miss earth since the diameter of earth is only 12,742 km, so it may be a little bit like getting caught in the middle of a 'shotgun blast' of space-lightning-filled hard radiation with much passing either side of the earth.