One occasionally does work and never gets paid for it. This is such a project.
A client of a client, wanted to incinerate some stuff. The possibility of having temperatures as high as 2500C was mentioned. Some elements that melt at temperatures below 2500C include: Al, Fe, Ni, Co, Ti, Zr. What the heck are we going to make this incinerator out of? That list pretty much covers all steels, the superalloys, and most of the exotic metals one runs across.
What you are left with, is some of the refractory metals, a few ceramics, and carbon-carbon. Tungsten (3410), Re (3180) Os (2700), Mo (2610). Molybdenum and Osmium probably can't be used, because they form volatile oxides. Osmium and Rhenium are both Platinum group metals, they are bound to be expensive. Tungsten gets used in quite a few high temperature things. It does tend to be kind of brittle; thermal shock, vibration, etc. can be a concern. Rhenium has some wonderful properties. Rhenium does get added to Tungsten alloys, to lend more toughness, thermal shock resistance, etc. to the alloy. Carbon-carbon can handle the temperature, but like Mo and Os, it has a problem with oxygen at high temperatures (coal burns). Some carbides and nitrides which will stand up to those temperatures, and have been used in mechanical applications. No matter what is used, it is likely that some kind of ceramic coating will need to be used, that has to protect the bulk from the chemical environment.
I would guess the cheapest solution is going to be Tungsten, with and alloy of Tungsten and a small amount of Rhenium being next. I have concerns about a Tungsten incinerator being able to go through restarts, thermal shock, and handling vibration. Rhenium probably has the mechanical properties, but at something like $50 per gram, it is going to be expensive. And, in applications like this, you probably have to convince the US government you aren't making rocket nozzles. I wouldn't be surprised if carbon-carbon didn't have similar export problems.
Sure, 2500C is hot. It pushes the limits on what a person can use to build things out of. The problem is probably better solved by designing it less like a conventional incinerator, and more like something else.
Rocket engines get up to that temperature, but they aren't made to last for anywhere near the time an incinerator would be. The application which probably lends the biggest hints for making these ultra high temperature devices are fusion reactors. They aren't talking 2500C, they are talking millions of degrees. And carbon-carbon is a prime candidate for first walls in a number of designs (or used to be). The idea is that the hot gas never really touches the structure. If the density of these very hot gases at the surface is low enough, there isn't enough heat flux to the surface to cause problems with melting.
Perhaps what you have is a cooling gas travelling close to the surfaces, to keep the hot gases away. You might be using microwave heating to get the temperatures as high as you want them, consequently your incinerator needs to take into consideration the properties of microwave cavities. I would anticipate you need to use pure oxygen instead of air, in order to avoid problems with nitrogen oxide generation. I have heard of plans to build a pure oxygen, gas turbine, where they would capture all of the exhaust (all carbon oxides) and pump it underground. No emissions.
I'm just a materials engineer (who has been a mechanic, partsman, equipment operator, welder, ...). I'm not a mechanical engineer hired to design this stuff. :-)