What kind of hazardous alien waste would tear apart society trying to obtain it?

extra-dimensional matter

Extra-dimensional matter can be used like anti-matter and be annihilated to produce energy. However, since atoms are on the order of 10^-10 meters a 10X10X10X10 cm cube (10 cm on each side) has a billion times more power than even normal anti-matter.

Why is it waste?

The beings that made it exist in 5 space so this 4 dimensional matter is considered an impurity and would have trouble moving things in 5 space at all. So it is harvested and ejected to ensure efficient and non-turbulent fuel flow.

They throw it out and eventually the unpowered cargo ship stabilizes to the most stable 5 d coordinates, (x,y,z,0,0) that is to say, three space with no variation in the 4th or 5th dimension, or our universe. The 5 dimensional beings don't care because they exist on a higher or lower position in the two other dimensions since there is more higher dimensional matter there.

So when 1000 tones of extra dimensional matter comes into orbit everyone realizes that there is currently more energy in that material than all the matter in all the fissile material on earth and that ownership of the craft means energy independence forever and unlimited military applications.

The pod could be organic. Organic ships (organisms tailored to be able to consume fuel, fly in space, ferry cargo or weapons and REPRODUCE, creating more ships out of raw material) should be the apex of any space civilization goal. Humans are unfamiliar with anything lose to it as of now, since our genetic researchers are yet to allow the creation of new species, rather than use science only to treat genetic ilnesses or in some cases improove already existing or endangered organisms. Whatever balance the power board have on the earth at your time would be chaotically shifted if one nation held the controll of this new piece. The pod, however could have been discarded for a reason... Maybe it's genetic code suffered a mutation, maybe the pod itself is an egg of such ship vessel... Maybe it is a cancer removed by the ships engeneers, who could have evolved to live within them like parasites live inside whales, completely depedent, and yet these aliens would have changed their biology to connect with the vessel by a cranial nerve, and without such, the ships would never accept being flew by humans or any other species. Or maybe the pod is infected by a bioweapon of the same civilization, engeneered to decompose metal alloys, kiling the aliens inside but leaving the ships almost intact. Lastly, if none o that works, maybe the pod came from other point in time. The last surviving record of a war fought between the current ruling nations, or a glimpse of a possible future, showing which of them shall rule upon the rest.

The totality of its properties are not known.

https://mondobizarrocinema.blogspot.com/2013/08/meeting-my-criterion-repo-man.html

In the movie Repo Man, there may be dead aliens in the trunk of a Chevy Impala. There is definitely something in the trunk. What, you have never watched Repo Man? Callow youth! Watch it! And do not click on links to learn more about the movie or go to you tube to see clips. Just watch it from the start.

In any case the nature of these aliens or whatever is in the trunk is not made clear. They can definitely do stuff, and do, in one of the weirdest and most awesome endings of any movie. Many parties want that car though their motives are not made clear.

So too in your story. The totality of what this alien material might be able to accomplish is not made clear. The conflict among people wanting the material is very clear. As to what could happen when someone finally does take possession of this material, I again refer you to the end of Repo Man.

Please see my answers to:

What would be the most expensive material to an intergalactic society?

And:

Could there exist a quantum virus which breaks someone's body down to fundamental particles?

In my answers I mention strangelets, a form of matter that works like katamaris. The TL;DR explanation of strangelets is this:

(...) a strangelet coming into contact with a lump of ordinary matter could convert the ordinary matter to strange matter. This "ice-nine"-like disaster scenario is as follows: one strangelet hits a nucleus, catalyzing its immediate conversion to strange matter. This liberates energy, producing a larger, more stable strangelet, which in turn hits another nucleus, catalyzing its conversion to strange matter. In the end, all the nuclei of all the atoms of Earth are converted, and Earth is reduced to a hot, large lump of strange matter.

Source: this Wikipedia article. The text in it has since changed, but the content is much still the same.

I think this is the best match for your requirement that it must be:

hazardous enough they launched it into space rather than incinerating/recycling it.

And:

(...) something that doesn't violate physics (no Pixie dust), ideally something that has been theorized to exist but not made/proven yet.

Also check this awesome Kurzgesagt video on how strangelets work: The Most Dangerous Stuff in the Universe - Strange Stars Explained.

Too long to fit into a comment, sorry. Nothing specific, just a very vague... feeling more than a concept.

Thinking about some folklore (golden antilope, or a food-brewing pot unstoppable once activated in greed)... Or some fantasy books, including Warhammer 40k... Our seemingly huge universe is a bubble inside some not merely hostile, but unimaginably and inconcilably alien real infinity.

In their scientific rush alien beings managed to burst a hole in the bubble skin and now this real-alien somethingness squeezes in. It is not bad or good, it is absolutely inconcilable with our existence. Like Ice-9 in space. They can not patch the hole or somehow push it back, they only managed somehow to split it to pieces and desperately throw it away, to keep their own star system from being immediately consumed. Consider it environmental disaster on a space opera scale.

Problem is, i can not come up with any plausible reason Earth nations would want it, short of some grotesque dark cultists.

Anti-matter blobs. Raw blobs. No "magnetic cradle" needed - that would not survive millions years in void anyway.

Why thrown away? Imagine those advanced aliens have some (Pixie dust, yes. But Pixie dust somewhere away outside of the "novel" plot) ultra-powerful energy generation process, that creates anti-matter in huge quantity as a waste. The quantity though has to be real huge, that annihilating it at place would not be just bonus energy, but more of destruction of alien's homeworld.

Or, maybe, it was not the energy they were creating, but rather they were creating the matter itself out of vacuum. World-building taken literarily. Not terraforming of pre-existing planets, but rather using energy to tear vacuum into matter and anti-matter, then using this matter to create new planets for their living (maybe even drifting planets, like in Niven's Ringworld) and safely throwing anti-matter to void infinities.

Why craved by humans? Well, exactly because it is an ultimate weapon! Capture it, manage to split and control (EM fields?) - and you can throw onto your enemy millions of bombs much more powerful (and much more environment-friendly!) than those vintage old school nukes. And if not used - mere blob has much more longeviety than hyper-complex tech of plutonium warhead. Just shield it from Solar wind and it can be stored for ages, as long as you can generate EM field anchoring those blobs in space.

Back to our XX century - our militaries tend to throw away spent tin cans, from meals or from cartidge boxes. And what use can a stone-age tribe make of metal, even if we ourself consider tin too soft to make blades of, but they have no any metal at all!

Or, if you do not want to go for grim comedy, then anti-matter might be craved mostly as ultra-rich power source, rather than weapon. Afterall all power sources have "dual use" (military and civilian depending how you arrange energy extraction). And instead of "kill them with fire" Earth nations would plan to destroy each other economically, leveraging ultra-cheap abundant energy.

Bonus point, if this "free energy" is not a "benevolent gift" or a random by-product, but a nefarious plot to destroy Solar system itself by our own hands. Which can be slowly revealed by charting those blobs and finding them not scattered all directions randomly, but precisely aimed at us, with regard to galaxy rotation. But here we tread into Asimov's "Gods themselves" :-)

UPD. @cowlinator brings a point of antimatter slowly "melting away" due to residual cosmic dust, star winds, maybe dark matter. I can not guesstimate matter density in interstellar voids. Nor the expected distance/ages of travelling in a novel. However this hints at one more plot device: approaching Sun the blob would be subject to quadratically increasing Solar wind flow, which would cause quickly increasing annihilation - light radiating, which would allow Earth to detect this object incoming, and maybe to try looking for halo of similar blobs drifting outside of Solar system too.

Metamaterials

The waste genuinely is just garbage.
The catch is that the civilisation that produced it does seriously cool things with material-science, and while most of their waste can be recycled, a small amount cannot.

Meta-Materials can have incredible properties, we manufacture them in various forms now in the 21st century so imagine the possibilities of a civilisation many centuries more advanced than us.

Matter-sieves
Light-bending fabric (invisibility cloaks)
laser-focusing blocks
Self-healing materials

The list goes on and on.

A lot of the applications are quite mundane industrial uses.
A light-bending foil/fabric would be excellent for optics and telescopes, matter-sieves can filter liquids and gases without significant energy input, self-healing plastics can repair damage done to them.
Being able to unravel how they work and how to manufacture them would potentially revolutionise a lot of industries.

The catch is that their complex structures are hard to break down and recycle, so when no longer needed, they're just trash.

Some metamaterials are quite dangerous to handle. for example a particularly effective matter-sieve might do unpleasant things to your hands if you touch it...
A Self-healing polymer might grow over anything it touches, trapping unfortunate astronauts inside it like a mosquito in amber.

One metamaterial being developed now mimics the behaviour of gecko feet. the idea is to allow people to climb vertical surfaces like a gecko. nearly microscopic hairs in the gecko's foot penetrate the material and provide enormous surface area to grip with compared to the overall size of the foot. with the quirk that it peels off like a sticker.
Now imagine gecko-foot material sheets in a trash-tangle...impossible to detach from anything that touches them. Any effort to escape just tangles the victim further and slowly draws them irretrievably into the mess...

Attempting to retrieve the materials from the space pod is highly dangerous because innocuous objects can have dramatically unexpected properties, some of which might prove lethal.

Aliens themselves.

The pod is not a garbage can, but rather a burrial tomb, sent in space either:

• as part of burrial ritual (in that case, the alien inside might be an important individual),
• or perhaps because the bodies inside are victims of unknown plague that broke out on a space ship (it was easier to dump them in space, than to incinerate them)

Why would humans fight over it? Could examination of the bodies bring any breaktrough in medicine or other fields? "Earth technology is several centuries advanced from the current time": Do humans believe they can analyze their brains and recover possible leftover memories? Could they even attempt to bring them back to life, at least partially or temporarilly?

Not sure if the answer suits your question, but in the second scenario it is hazardous alien waste, possibly hazardous even to humans, and it's not a living waste.

The one who controls the spice, controls the galaxy

You could make it add a temporary psychic link to whoever consumes it due to the whole waste cloud being quantum entangled in a unknown way.

It could just be like Oxigen. A powerful chemical that can power creatures and machines, but is a waste product of other organisms (plants for the oxigen RL example). Check the periodic table for the theoretical highest undiscovered element. It could improve the society just by being a powerful energy source that breaks down in other useful elements during the power extraction. Having a better way of life is always a good way to start fighting between peoples, as no one wants to be left out.

It could possibly be used to enhance people, making them smarter, stronger and longer young. Think of people learning languages like when they were a baby, or growing muscles to the max fitness while just resting. Or giving people the power to make their conscious mind tap into the vast power of the unconscious, able to hold so much information that science jumps ahead by leaps and bounds.

Edit: Or just a drug. The drug filtering into the atmosphere is already enough for everyone to start craving the real stuff.