Incinerators are used to reduce the amount of material that needs to be put in a landfill. Basically, the trash is burned, and energy may be recovered from the combustion process to produce electricty. The first waste-to-energy facility was operational in Hamburg, Germany in 1895. In the end, after the waste materials are burned, the leftover ash, significantly smaller in volume compared to the original volume of waste, can be dumped in a landfill. This ash may contain heavy metals and toxic compounds. If that is not concerning enough, of course burning the trash creates pollutant emissions. There are the usual suspects, carbon dioxide and oxides of nitrogen, but there are some smaller, toxic (scary toxic) compounds, like dioxins and furans. One of the most famous uses of dioxins was Agent Orange (don't click on the link if you are queasy), which was sprayed on fields during the Vietnam War as a part of herbicidal warfare methods. Dioxins are generally formed in the smokestacks of incinerator facilities, where temperatures drop to less than 650 degrees F. There are several methods of cleaning this emissions of heavy metals and dioxins, but I haven't found anything that talks about how these concentrates of dioxins are dealt with.
It is interesting that the argument for either inaction or lessened regulations on action due to large uncertainties in our understanding is not only used in the climate change debate. Apparently, the World Health Organisation says that the impacts of dioxins on human health are uncertain, and therefore "exposure should be limited." Clearly, Agent Orange caused birth defects, but we still allow dioxins to be released into our air and water. Indeed, are we willing to curb our use of products if dioxins are formed when they are incinerated after we throw them away?