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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.
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.
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.
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.
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.
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.
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.
Just to add to this: rockets use a lot of fuel. And with "a lot" I mean that a rocket typically is 90% fuel, 7% rocket, 3% cargo (my numbers may be off a bit, but not by much). The further you want to go the more fuel you need, the heavier you get,the more fuel you need, and so on.
So to move out 100 tonnes of cargo, were going to waste, say 3000 tonnes of fuel, and that is just to get it into orbit. Getting enough speed to get it to the sun would probably literally require exponential amounts of extra fuel, which would require extra fuel rockets to come up, rockets just carrying "a little bit of" fuel for another rocket.
Then on going to the sun: the earth moves at about 30km/s around the sun. To cancel that out, you'd need a rocket capable of reaching 30km/s, which we currently -afaik- cannot. We can't get rocket engines that can eject the burning gasses out at those soeeds, hence we can't reach that speed, hence we can't cancel out 30km/s. We'd need entirely new technology to be able to do that
To get into the sun, we'd probably want to fuel the rockets in space using reaction material mined in space (from the moon or an asteroid). That would more or less eliminate the problem you're talking about, which is why I kind of skipped over that in my comment. But you're right; this is one of a million things that makes space travel hard and expensive.
We can get up to any speed with enough time and fuel. The trash rockets would just need to get into a solar orbit, and then burn retrograde for a fairly long while. Or if you add a gravity assist in, this is doable today; the Parker Solar Probe got to (and indeed beyond) that speed, for instance. It's easier and quicker when there aren't squishy people aboard (we don't tend to like acceleration much higher than 9.8m/s², for instance).
Oh, also: I don't think it's a stupid question. It's a fun question. It might not be a workable plan, but I love thinking about this stuff.