I have built an automatic device for incinerating refuse with magma in my current fortress, Pagedslipped.
The device is built around a single tile stockpile which accepts remains, body parts, rotten skins and spoiled food, and other undesirable materials. Access to the refuse stockpile is through a short corridor containing pressure plates and hatches. The first pressure plate is a safety switch which triggers only for friendly creatures of less than size six. This is linked to a hatch over an escapable pit immediately following it, and serves as a safety device to keep children and pets from following dwarves through the device.
Immediately before the refuse pile is a second pressure plate which triggers for adult dwarves. This pressure plate opens a hatch in the entrance corridor, blocking the dwarf from exiting the way he came in, and opens a door to a different corridor leading out of the chamber. When a dwarf drops off a chunk of refuse in the pile, he will then find the way he came in blocked, and will have to take a separate path out.
The path out has a pressure plate which opens a floor hatch, holding a tile full of magma, above the refuse pile. It also triggers a floodgate located on the exit of a pump which removes magma from the pile. The hatch triggers immediately, dropping the magma on the pile. A hundred ticks later the floodgate opens and the magma is removed from the pile and pumped back onto the top of the floor hatch. This is enough time for the refuse to catch on fire. Once it has burned away, the device is clear and ready to use again.
This device automatically incinerates refuse, and requires no manual activation on my part at all. It's also so far failed to accidentally set any dwarves or animals on fire.