In Justice League Dark, we see the following, what looks very much like the death of Swamp Thing:
Is Swamp Thing officially dead in canon now?
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Swamp Thing may have been "killed" in the JLD movie, but it is unlikely to be permanent due to Swamp Things elemental nature.
Answering the question if Swamp Thing was killed in the Justice League Dark animated film is difficult without knowing the direction of the future story arc, if any. I have tried searching for information about the story arc without much luck.
That being said, there are a some things I can point out as to why it is unlikely that Swamp Thing would be permanently killed.
The first is the background of Swamp Thing :
The creature, called Swamp Thing, was originally conceived as Alec Holland mutating into a vegetable-like creature, a "muck-encrusted mockery of a man". However, under writer Alan Moore, Swamp Thing was reinvented as an elemental entity created upon the death of Alec Holland, having somehow absorbed Holland's memory and personality into itself. He is described as "a plant that thought it was Alec Holland, a plant that was trying its level best to be Alec Holland" Saga of the Swamp Thing #21, p.22
It is this version of the character that is present in the Hellblazer series featuring Constantine. The following image shows an examples of the dichotomy between the elemental creature and the human analogue that it is associated with - i.e Alec Holland. The image shows Constatine describing an interaction with the "green" itself, noting that it is nature in its raw form and that Holland is not there.
Making the assumption that the character in JLD is an extension of this same character, the removal of the "humanity" associated with Swamp Thing (the body being removed) would not kill Swamp Thing, but would likely cause interest in human affairs to diminish; the indifference of nature. It is possible to conclude that a future iteration of Swamp Thing could possibly not have the personality associated with Alec Holland.