A lot of the metals end up getting spewed into the atmosphere. Lovely, eh? And some of the metals might be radioactive (incinerated smoke detectors for one). And doesn't that sound even better!
The ash primarily consists of carbon followed by nitrogen and maybe some other heavier elements. It is also possible that asbestos may be left behind. But it is also possible that the incinerator could be running hot enough to vaporize/burn the remaining materials. Virtually all the chemical bonds are broken and the elements get oxidized and emitted as some COx, NOx, SOx, or something to that effect. If there are any solids remaining behind, they get collected. Depending on their composition, they might get collected and disposed of as hazardous waste. But if it's primarily carbon (as mentioned before), then it might just get disposed of as regular trash.
Also, incinerators would likely have scrubbers (filters) that collect the NOx's and other pollutants, including the heavy (and radioactive!) trace metals/elements. These scrubbers undoubtedly get disposed of as hazardous waste since they are collecting the nasty parts of the smoke fumes.
I'm not 100% sure if there are also filters for other types of pollutants, like chlorinated and bromated gases, benzene, chloroflurocarbons, polycarbonate biphenols (PCB's), and other types of nasty chemicals that are environmentally bad. I would assume so, but I havn't done such research of that nor have I heard of such things. This is probably one big reason why incinerators are slowly going away because dealing w/ these kinds of emissions are difficult, expensive, or perhaps not possible in effective, practical ways w/ current technologies.
Don't live within a few miles of an incinerator, they *will* effect your health in a bad way. There are coal-fired power plants in a couple neighborhoods of Chicago and the incidents of respiratory ailments among the public, especially the children, are triple or higher compared to the persons living farther away. And this is a powerplant... The incinerators are burning everything and almost anything imaginable. Find out if there are any in your neck of the woods and get out if there are.
BA in Physics, lived in Chicago (city) for 7+ years. Worked as an industrial hygienist for 5 years including site assessments for environmental hazards.