For medical waste, this might do, but then a good stove might do too.
For nuclear waste, that's the last thing you want to do with it! The radioactive nature of the material is not altered by the "low" temperatures of a few thousand degrees, which only affect the chemical state and not the nuclear state. You'd need something much hotter to "incinerate" nuclear waste, of the order of a few million degrees at least (like the core of the sun).
The only thing a volcano would do, is to spread out quite effectively the radioactive material in the biosphere. Kind of "superchernobyl" style of exercise
The best thing you can do with nuclear waste is to keep it shielded away from the biosphere for a few hundreds / few thousands of years: it decays by itself. You want, at all price, to avoid it being spread massively around too early. That's why deep geological burrying is the "standard" solution to the nuclear waste problem. You want to keep it there for a few hundreds or a few thousands of years. Think of gas an oil and coal who have been confined for hundreds of millions of years in geological confines. A few thousand years shouldn't be such a problem. The last thing you want is to have it in an active volcano.
BTW, the fact that it decays by itself is one of the favorable properties of nuclear waste, which is not the case with certain kinds of chemical waste for instance. Heavy metals remain heavy metals for ever.
The next best thing to do with it, is to transform it actively. Although this can in principle be done, it is technologically an enormous challenge.
Sending nuclear waste in space is an idea that comes up regularly, but is basically flawed: the dangers of a failed launch are too great, the price is too high, etc... compared to the "standard" solution of deep geological burrying.