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Prohibitively expensive.
First the cleanup is gonna take forever and cost billions.
Then building a rocket is gonna be even more billions and time.
And then actually shooting something into the sun is harder than just blasting it out of the solar system.
You could save a bit by shooting it into another star, and not our own. But you still gotta clean it up and make a rocket. I don't think we have even launched a rocket that big or heavy ever. It may require multiple rockets. Planet Express barely was able to make it happen, and they are in the future, only needed to clean NYC, and is also from a cartoon.
Why is this true? Wouldn`t gravity do most of the work if we just kinda shove it in that direction?
The earth is traveling around the sun at about 67000mph (29,722 meters per second, the unit of measurement I'll use from here on our for consistently) that means to fall into the sun (and this is once you've already expended a ton of Delta-V (delta-V being a count of meters per second in change to orbit your craft needs to make/can make) escaping the Earth's gravitational influence) you'd have to slow down a significant portion (about 24,000 meters per second specifically) of that 29,722 meters per second that you're hurtling through space at.
It takes so much energy to try to crash a craft into the sun it's literally cheaper (only costing about 8,800 m/s of Delta-V, compared to about 24,000 m/s of Delta-V) to fly the craft very very far away, such as to the edge of the solar system, then zero out the angular velocity so it effectively falls into the sun, than it is to fly directly to the sun. This tactic also enables one to use another planets gravitational influence to "gravity turn" and save on fuel, but it's still horrendously expensive to get even a small craft weighing a fraction of a ton from the surface of earth out to the edge of the solar system to begin with.
Rockets face a significant challenge in that in order to reach orbit they need a large amount of energy, sources from a large amount of fuel. To get 1 ton of payload to orbit it needs an amount of fuel which adds additional weight which then requires additional fuel to lift the mass of the fuel. Because of this it takes about 100kg of fuel to get 1kg to orbit
In short, I highly recommend spending a few days playing Kerbal Space Program to learn far more than will fit in a single comment about orbital dynamics. That game is amazing at teaching basic concepts of orbital dynamics and the incredible challenges space programs face in just getting payloads to orbit let alone incredible feats like interplanetary travel or interstellar travel
Because if you launch something from Earth, you inherit the Earth's orbital speed around the Sun. At that point, whatever you launched will just continue to orbit the Sun. It takes less energy to accelerate to a solar system exit trajectory than it does to scrub off all of the excess velocity and end up on a trajectory that intersects the Sun.
But does it matter what speed the garbage is going at when it hits the sun?
No, but it's going too fast sideways. It would miss the sun. You need to slow it down by the same apeed that Earth is moving, stopping its sideways motion and letting it drop into the sun.
Edit: I like making diagrams. Red is the trajectory you're expecting. Blue is the Earth's motion, which adds to that red arrow. Purple is the resulting actual movement of the trash rocket.
But do you need to slow it down all the way? Can't you just slow it down enough to get the ball in an elliptical orbit where the trash ball gets very close to the ball of plasma?
Space is big. It's so big that our tiny ape brains have a hard time conceiving of how big it is. The sun is actually (despite it's size) a relatively small target and is very very far away. Now the more delta-V you burn to slow the trash down the smaller its orbit around the sun will be. But that orbit starts enormous. So to get that purple line near the sun you do need to slow down almost the whole way, just to get it close.
The problem is slowing it down to any speed that would end up with it dropping into the sun is going to take more effort and be more difficult than firing it out of the solar system. It isn’t practical.
Yes and no. The gravity of the sun will attract the rocket, but there are other things out in space besides the sun.
The problem then is other planets will start whipping the garbage rocket around who knows where. Could even come back around and smash into earth. Same problem with the sun, actually. It's quite hard to hit something that's that big when we're this far away. If you miss even a fraction of a decimal of a degree, the trash rocket will swing around and you're back to planetary hot potato.
It's easier to sling the rocket past the south or north pole at a right angle to the solar plane. Up or down it'll either keep going till it's another suns problem or it joins the Oort cloud, which is kinda like a giant trash dump for everything that didn't make it into our solar system when the sun formed.