Flammable artifacts will catch fire, but as artifacts cannot be destroyed, they will burn forever. That's what's happening to you.
I don't think that smoke will harm anyone, although it gives dwarves unhappy thoughts. It doesn't do anything else, as far as I know. Since the artifact is still actually burning, you can possibly drop it into a pit and dump burnable refuse onto it, creating an incinerator. Using the smoke for aesthetic purposes is not out of the question, of course.
You could use it as bait for thieves. Put it out in the open (beware of hauling it, since it's on fire. You may have to sacrifice a few dwarves for this purpose) and kobolds will try to steal it because of its high value, but catch fire and die before making any real progress.
Also, see
this page for a trap involving burning artifacts.
The outside of the fort is now a field of scorched earth and feather trees (which seem to survive fire quite nicely). Two puppies are permanently flashing red but don't seem to have suffered any wounds (a bug I'm assuming).
Trees will never burn, oddly. Also, the puppies are flashing red because they've suffered blood loss. If you look at their wound status (v -> w), they should be listed as "pale" or "faint". They'll recover after a while.