First - The major problem with trash isn't the getting rid of it part, it's the gathering it up part. If we could do that, it wouldn't be a problem.

Second - Launching things on a rocket is kinda dangerous still, there's a risk the rocket will blow up on launch, scattering the material across a large area. This is a big reason why things like nuclear waste is a problem to transport in general, much less flying it somewhere.

Third - Launching something into the SUN is really hard, it would be easier to send something out of the solar system than back into the sun.
https://van.physics.illinois.edu/ask/listing/43694

Fourth - Someday we'll figure out a use for everything, wall-e style. If we dump everything into a centralized landfill, we'll eventually be able to collect/sort/recycle it into something useful. Throwing it into the sun (or off-planet) would make that stuff unavailable forever.

Finally - Throwing stuff into the sun would actually get rid of it forever, yes. It would be completely decomposed into the atoms it was made from. If we threw ENOUGH heavy metals into the sun, we could actually poison the sun making it not able to fuse hydrogen anymore, but even if we threw the entire earth into the sun, it wouldn't be enough.

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.

Because incineration or proper disposal is not the problem. Gathering and segregation is. Plus, launching that sort of payload is going to be insanely costly.

The sheer volume is manageable as it currentlyis, but it's spread out so much that collecting it properly is going to take a lot of time an effort.

The Great Pacific Garbage Patch is a bit of a misnomer, as it's more of a vague area in which trash tends to collect. It's not like an actual continuous patch that you can easily attack with a net.

1. Just gathering all the trash would be tricky (and, rocket aside, if we could do it easily, we'd probably have done it already; and just put it in a big garbage dump or something). Think about a swimming pool with a bunch of fallen leaves in it; it's moving around constantly, and if you swim toward one it'll kind of move away from you or break up when you try to pull it out.

2. Ok, let's handwave getting the trash out of the ocean. It's probably a solvable problem. First we need to sort it; all of the recyclables need to stay and be recycled, because we still need that material and because we need to reduce the weight. Compostable stuff can probably also just stay and be composted. Corrosive stuff probably shouldn't go on a rocket. All of the wet trash (it came from the ocean, it's all wet) needs to be dried out first; partially because we need the water, and partially because water is really heavy. And once we've done all of that...well, trying to figure out something productive to do with that big pile of dry trash is almost certainly going to be cheaper than launching it into space.

3. Ok, let's handwave that problem too; let's imagine we're just going to grab it out of the water, compress it, and get it onto a rocket. Except we're going to need a whole lot more than one rocket; a decent guess says that we've launched 18,003,266 kg into space ever—over our entire history in space—but the Pacific Garbage Patch alone is estimated to be at least 45,000,000 kg, meaning we'd need to launch more than twice the number of rockets we've ever launched before. More than 60,000 rockets have been launched since 1957, so that's substantial. It would take a while; even if we turned the entire space industry's output toward the project, they're "only" launching about 1,000 rockets a year nowadays, so it'd take at least 120 years of NASA, SpaceX, Blue Origin, Roscosmos, the ESA, the Chinese Space Agency, etc. doing nothing but trash full-time.

4. Ok fine. Again, we're handwaving; let's imagine we have everything loaded up on rockets on the launch pad. Just getting it into orbit is tough for the simple reason that we have to take not just the payload (the trash) but also the fuel we need to get it there, and to get that fuel off the ground we need fuel, and to get that fuel off the ground, we need— you get the picture. The Tsiolkovsky equations govern how much, and thankfully the number isn't exponential. But we will still need a lot of rocket fuel. Good thing we're devoting the entire space industry's output toward this for the next 120 years.

5. Now it's all in space. Great! That was actually the easy part. We could just leave it in orbit around Earth; that would be a really really bad idea for a lot of reasons (but it's what we're already doing with our space junk, so...), and you said "into the sun," so let's talk about getting it there. Believe it or not, getting it into the sun is actually way harder than getting it out of the solar system entirely. If you were on a rocket, and you pointed it toward the sun, and you burned and burned and burned and burned until you ran out of fuel, you would counterintuitively end up somewhere out past the Earth's orbit on the other side of the sun. This is because you have to actually cancel out your (very fast) orbital rotation, which you inherited from the Earth when you launched, before you can get pulled into the sun; otherwise you just end up going around the sun in a very elliptical orbit. It takes a lot of fuel to cancel out Earth's substantial orbital rotation. So we have to get that up there too.

6. The good news is, once you get it to the sun, you're good. It won't cause any noticeable change to the sun (the entire Earth could fall into the sun and it wouldn't care). And while the trash would initially melt and then burn due to all the heat, smoke is entirely a product of atmosphere and gravity; so no smoke would be generated and it would not make it back to Earth. But once all the ash made it to the sun, it wouldn't continue burning per se; the sun doesn't produce heat by burning, but by fusing lighter elements into heavier ones. The Garbage Patch is mostly plastic, so carbon polymers. But the sun isn't big enough to fuse carbon into magnesium, which means all of those carbon atoms would just kinda...sink into the sun, hanging out under all the hydrogen and helium and lithium and beryllium and boron, but on top of the nitrogen and oxygen and such, for the next ten billion years until the sun turns into a red giant. Then, the sun will expand outward, potentially to engulf the Earth's orbit; at which point it will reclaim all the atoms of the trash we didn't send up there.

7. Eventually, after a bunch of different cycles and drama, the constituent atoms of our trash and everything else would become part of the white dwarf that our sun will become; a small, slowly-cooling stellar remnant. After that...we don't know! The time it takes for a white dwarf to cool completely is longer than the life of the universe so far, so we have to speculate. It's possible that the remnants of our sun and our trash and everything else might end up becoming a black dwarf, which might look like a shiny spherical mirror the size of the Earth.

All of that seems like a lot of work. I think we should try something else.

It costs about $10,000 US to get a kilo of payload as far as Low Earth Orbit. I'm not sure this is going to scale up.

The short answer is just that doing so would be ridiculously difficult and expensive. Funnily enough, "launch it into the sun" is actually the easy part at this point. If we could collect all of the ocean's trash, we probably would have done so and compressed it by now.

Launching stuff into the sun takes a shitton of delta-V. We should just launch it into the moon.

I'm fairly certain there's a Futurama episode on this topic

Gathering all the trash to launch it into the sun isn't easy, as many comments have pointed out. Not only do you have to counteract the velocity of Earth, but I'd expect you'd need a way to keep them alive on the trip there as well. I mean, I'm assuming you want them to be cognizant until the end, yeah?

Even if you could do this, it would be more effective to just do the "collect all the garbage" part and then store it in a heavily lined container forever.

I think you wildly underestimate the amount of trash we're be talking about here. This wouldn't be a rocket, this would be thousands, or hundreds of thousands of rockets. And that's just the start.

putting anything on orbit costs A LOT, per kg. Not worth it

Just drill a hole to the core of the earth and dump it there. And you would just put a restart on all the materials.

Orbital mechanics makes launching stuff at the sun extremely difficult.

The earth has a gigantic a molten layer under our feet, and we couldn't even dump it down there. Too expensive and difficult.

Long term, my guess is engineered super bacteria and/or robotics may clean up the trash in the future, if we don't extinct ourselves first.

This is an incredibly stupid question.

OP:

And the one time the rocket goes kablooey on its way up, everyone down the flight path will get a shower of used hypodermic needles, disposable vapes, and old appliances.

You got it: it's expensive/ "not financially viable," so it won't happen

First steep, gather all the ocean's trash.

Good luck with that !

I also have this incredibly stupid idea to build a very long pipe that goes all the way outside earth's gravity pull, then launch all the garbage through it with a mechanism similar to a railgun. It doesn't have to be thrown directly at the sun, just enough to launch stuff out of orbit.

"Futurama – The Great Garbage Crisis of 2000"