Biohazard waste (removed body parts and those disposable tools that come in contact with your blood, for instance) are destroyed in a way that is designed to make sure that no living cells could remain, such as dissolving in chemicals or burning to ash in an incinerator.
I would like to point out that body parts that are removed are not necessarily unnecessary- even the appendix, the archetypical useless organ, is currently thought to provide some function (providing a safe store of positive bacteria to re-populate the intestines after something like diarrhea flushes them out)
Pathologists' Assistant and med tech here. When you get a body part removed it will come to the anatomic pathology lab or clinical lab. The clinical lab deals with liquids (blood, urine, spinal fluid, etc.) and microbiology while the anatomic lab is exactly that: for anatomic parts!
So if you got your appendix removed it will go the anatomic pathology lab to be "grossed" (measured, described) and sectioned. What we do to the appendix depends on what it was taken out for.
No matter what we will snip off the entire distal tip (the end opposite the side that attaches to the cecum) as that can harbor a kind of tumor called a carcinoid tumor. We will then serially section the appendix looking for any abnormalities. What does the lumen contain? Fecal material, fecoliths, blood, pus, nothing? How does the serosa (the outside of the appendix) look? Is it dull and not shiny? Does it have purulent exudate on it or the attached fat (mesoappendix)? Is it torn or is there a perforation? Finally, we look at how thick the wall of the appendix is (they are usually about 0.2 cm thick).
A normal gross dictation of your presumably normal appendix that may have been incidentally taken out will look something like this:
Received fresh labeled with the patient's name, medical record number, and additionally labeled "appendix" is a 6.5 cm in length by 0.5 cm in diameter appendix with 2.0 cm of attached mesoappendix. The serosa is pink-tan, smooth, and glistening. The specimen is serially sectioned to reveal a 0.2 cm lumen containing blood and a 0.2 cm thick wall. Representative sections are submitted as follows:
1A = entire distal tip bisected longitudinally
1B = representative cross sections including inked proximal margin
Edit: So if you have appendicitis I will look for purulent exudate on the serosa, in the appendix itself, and will make sure there isn't a perforation.
Edit 2: The sections I submit in cassettes are then processed and cut by histotechs. These small (think micron size sections) are then stained (usually with hematoxylin and eosin) and the pathologist will examine the slide under the scope. They perform the final diagnosis; in the case of appendicitis looking for neutrophils inside the lumen.
The vet hospital of the local university has a chemical vat that they dissolve remains in, should the need arise. From what I understand they only do this if there is some reason which the animal cannot be disposed of by other means. Communicable animal diseases come to mind.
I've got a story about the old methods of disposal. For reference, I am an environmental geologist and was working near Flint, MI a few years ago. Had to do a subsurface investigation at one of the vacant hospitals because for 30+ years they dumped all biowaste into an open pit west of the hospital. They filled in the "skin pit" (what my coworkers and I dubbed the dump) in the late 1980s with building debris from an old on-site church, and then paved over it for extra parking. We drilled the pit and found significant contamination, including formaldehyde, in the groundwater. All of the houses down-gradient of the hospital had to connect to city water. Pretty gross considering how long they were drinking that.
Your appendix would go to the pathologist and get fixed in formalin initially so it could then be examined under a microscope to assess the cause. A very small number have tumours. In general most excised organs get sent to pathology before disposal.
Had my wisdom teeth pulled and wanted them. Was told no it's biohazard waste. How did they go from my extra teeth to biohazard in 30 minutes. No one could explain. Did not get them.
It's special Medical Waste. Yes, they throw them out, like the bloody sponges, unless they need to go to the lab for a biopsy.
It varies drastically across the world, but medical waste (including removed organs, but also many other forms of medical waste ranging from bloody bandages to used needles) is usually incinerated. https://en.wikipedia.org/wiki/Biomedical_waste
It's not the biggest body part, but I worked in a lab that collected and used foreskins (from circumcisions). Infants tissues can be harvested for stem cells and it turns out cutting off other bits of the baby is commonly considered unethical.
For the record even though this doesn't really pertain to your question: your appendix is [not totally useless](https://www.sciencedaily.com/releases/2007/10/071008102334.htm) as once thought. It produces gut flora, which is to say bacteria in your digestive tract that help digest food. You can live without it, but it's not useless.