Two issues only;

1. Cost (specifically launch costs), it has to be cheaper than processing (or storing ) the waste in question on Earth. This is unlikely at least for the foreseeable future.

2. Launch safety; Technology fails, some types more often than others. Depending on how toxic/dangerous the material you are launching is (and how much of it there is) you need rockets/rail guns etc with an extremely high level of reliability. Which, as noted above is expensive.

Imagine every 1000th launch failing and dispersing garbage all over the launch area or populating near Earth orbit with fast moving projectiles.

Use Jupiter instead.

• Jupiter is much more easy to reach with a rocket

• Jupiter does not have a solar wind that might carry your waste back to you

• Jupiter is a gas giant without strong convection

The last point is relevant because it means that your heavy nuclear waste will simply sink into the planet and remain there. The perfect end storage solution.

If you are already committed to dumping your waste in space, Jupiter is the place to send it. It still needs a lot of effort to reach, but there is no safer and easier destination in space than that.

You can toss anything into the sun, at 2.6 Million degrees in the convection zone anything tossed inside will be totally incinerate, course that'll happen before it reaches the photosphere anyway

As others have said, the sun can take anything. However, dumping waste in the Sun requires us to cause a lot of environmental devastation on Earth in order to collect the resources for this enterprise; Since not of all the waste will go to the Sun, this process is not sustainable.

In order to make this work, we would need some magical technology (in the Clarkean sense: something so advanced we could not comprehend it today) that would zero out the costs of sending stuff to the Sun.

Jupiter, while not a star, sucks up asteroids and such all the time with no major impact. Considering any trash heap you might drop into the Sun is basically a glorified asteroid and the Sun is about 1000x more massive than Jupiter, you probably won't even notice a difference.

One aspect I don't see addressed in the answers here with regard to safety is the mass from Earth you lose by doing this. Yes, garbage -- and especially nuclear waste -- is presently a challenge for us now, but destroying it by sending it into the sun negates the potential for future technology to turn such garbage into useful material.

If we were to send all of our garbage to the sun, the Earth's mass would gradually decrease as we exhaust natural resources assembling materials that are used, discarded, and fully destroyed/burnt. So, in essence, we trade the problem of where to store garbage we want to discard for a problem of losing mass that could be useful in the future. Do it over several centuries and I'd imagine the mass lost would be sizable.

If one is ready to spend THAT MUCH on disposing off some kind of waste, this begs the question how much dangerous is the waste in the first place.

As of now, the most dangerous waste we produce is the spent nuclear fuel and/or its reprocessing tails.

Probably equally bad are some highly reactive or poisonous chemicals.

In fact, we already have a means of managing it safely on Earth. Not that we always do, but it will still be cheaper (and pretty much more sustainable) than using rockets to move it to space (and even larger rockets to get it to the Sun). Rockets have their own eco implications, we are okay-ish with them because we don't use them much (compared to, say, automobiles).

Since you are talking about other stars - depending on your star and other planets in your system, it may be, unlike our Sun, easier to reach the star rather than the deep space.

What's more, in most cases it is generally better to keep some form of exotic waste until it happens to be a resource. The history of mining industry knows a lot of cases when a bug pile of waste that no one wants suddenly becomes a valuable resource and people start reprocessing it.

Then again, the technology moves forward and we constantly invent more and more unpleasant forms of waste. How about https://en.wikipedia.org/wiki/Strange_matter ? If it is able to convert the ordinary matter into itself, it will be equally bad (or even worse) at the Sun - instead of killing a single planet it will kill the solar system (probably later, so open to smuggling and corruption).

"Waste" is Fuel

Radioactive waste is fuel. We just call it "waste" because we have primitive thermal neutron reactors which are very picky divas and demand fairly enriched U235 (well, if you count 3% as "fairly enriched", which it is, compared to the natural composition). In fact, current LWR reactor technology only "burns" about 1% of the uranium in a fuel rod, leaving 99% of the fissionable material. On the other hand, fast neutron reactors can "breed" more fuel than they consume by transmuting the normally useless U238 into P239. They can also transmute thorium, making it a nuclear fuel source. In general, fast neutron reactors can "burn" just about all the long-lived radioisotopes which occur or result from commercial fission reactions. The only question is how much energy you want to extract.

Breeding, Burning, Garbage Disposal

A fast breeder reactor can produce fuel while also producing useful energy. India's nuclear program is predicated on the idea that you can use a small stockpile of enriched uranium to bootstrap an entire nuclear industry that ultimately runs on thorium (of which India has abundant supplies). Whereas, thermal neutron reactors exclusively "burn" the easy fuel. If you are willing to forego some of your power generating capacity, you can essentially make a fast breeder reactor burn all its fuel, which will get rid of more than 99% of the nuclear waste you can produce.

In this mode, the reactor will not be as efficient, because unproductive daughter products will build up, poisoning the chain reactions. Even so, the constant bombardment of neutrons will force the daughter products to fission down to a small handful of products which are far more easily managed.

Note that some of the fission products are commercially valuable, such as Krypton-85 (no, not to defeat Superman), Strontium-90, and Caesium-137.

Economically it is less expensive to shoot a rocket into deep space and more expensive to send it to the sun.

It is even less expensive to crack the waste down and recycle it, by far. For the billions you have to invest in a repeating sunward launch, you can research, engineer and build a transmutation reactor instead and gain lots of value from the waste.

Finally, when you start to put strange things into the sun, it's like signalling. Under the assumption that there are aliens and they watch with good spectroscopes, they could maybe see it. So if your universe is hostile, that can be a danger in itself. I don't know if that signal is stronger than our radio emission, though. Probably not.

My main point is, sending stuff to the sun is just super duper expensive. Basically everything else is cheaper, even sending stuff away from the sun.

Dumping radioactive waste into space was actually considered in the real world

The idea of launching hazardous waste into the sun is completely absurd on so many levels. Or is it? Let's look at the real world studies that seriously considered it. HDE 226868 has a terrific answer about the physics involved, so I'll stick to the history.

The 1978 NASA report Nuclear Waste Disposal in Space approved of the idea dumping waste into space: "Of the five space destinations considered, the lunar surface and solar orbit options are the most attractive..." The report declines to support solar impact disposal of nuclear waste only because it was beyond the rocket power available at the time.

A 1981 report went so far as to rank the different ways of hurling nuclear waste into space. Sending the waste directly into the sun was ranked last.

The same report illustrated the general procedure for launching radioactive waste into space.

Now to address your question. The idea of shooting waste into the sun has appeared again and again over time, and a study from 2011 answers exactly what you want to know:

The underlying principle here is that all matter caught in the sun’s gravity will lose its structural integrity due to the stress of gravitational forces and “break up” before reaching the sun. Moreover, high temperatures will incinerate and completely consume all matter prior to its reaching the sun’s corona. Specifically, as matter heats up, it expands beyond its structural integrity, and the heat energy encountered causes molecular bonds to break. Even the atomic integrity of elements of atomic number above two (i.e., helium) does not exist within the sun. Essentially, the intense heat renders such elements into their composite subatomic particles (e.g., electrons, protons, neutrons, etc.). Thus, the radioactive nuclear waste never impacts the sun, having no effect upon its “ecosystem,” and therefore cannot “damage” the sun.

If you could get the materials safely away from Earth and actually dump them in the Sun, no, there would be no danger.

The Sun is approximately 330,000 times the mass of Earth. To substantially change its evolution, you'd likely to need add - and I'm really just making an educated guess here - something on the order of $sim0.01M_{odot}$, where $M_{odot}$ is the astronomer's notation for solar masses. Clearly, that's impossible to attain using just the material on Earth. Each year, the Sun loses roughly $10^{-13}M_{odot}$, which comes out to around $10^{14}$ tons. However, humans produce about $10^9$ tons of trash each year, meaning that we'd have to dump about 100 centuries' worth of trash into the Sun to simply negate the mass it loses each year.

It's also unlikely that any dangerous materials would come back to haunt us. While the Sun does occasionally display bouts of activity, such as coronal mass ejections and solar flares, the material is dissipated throughout space, and the odds of the ejected matter containing our waste in significant quantities are extremely low. Odds are good that it will simply mix throughout the solar photosphere (in its now-ionized form) and never come close to Earth. The solar wind, which is fairly constant, is also not a substantial risk for essentially the same reason.

If the Earth were to be engulfed by the Sun, it likely wouldn't cause much of a change to the Sun's evolutionary track. It would, however, be noticeable, causing changes to the composition of the solar photosphere. We've seen this in, among other cases, the star HD 240430, which seems to have eaten several terrestrial planets, leading to a significant enrichment of photospheric metals. However, as this is only a change on the surface, its unlikely that the matter will affect the behavior of the stellar core and inner layers, and by extension it will not significantly influence its evolution.

If you really want to run the numbers: The main sequence lifetime of a star is approximately $$tau=10^{10}left(frac{M}{M_{odot}}right)^{-2.5};text{years}$$ where $M$ is the mass of the star and $M_{odot}$ is one stellar mass. It turns out that if you compare the cases where $M=M_{odot}$ and $M=M_{odot}+M_{oplus}$ (where $M_{oplus}$ is one Earth mass), i.e. the current lifetime and the lifetime if the Sun swallows the Earth, you get a difference of something like 75,000 years. For a star that should live about 10 billion years before leaving the main sequence, that's not much.

Now, as I alluded to at the beginning, we do have to deal a side effect of the problem of getting matter to the Sun. From an orbital mechanics perspective, I'd argue that it's not a major issue; it's not easy to get to the Sun, but it's far from impossible. However, if the rocket explodes - either on the launch pad or in the air - that waste would be spread out across the surface, and the resulting fallout could cost many lives.

Is this idea safe? Once you're out of Earth orbit, yes. Before that . . . maybe not.

The sun currently contains about a Jupiter mass of iron, left over from its formation.

Putting the entire Earth into the sun would not affect its lifetime significantly.