It is undoubtedly true, as Hot Licks's answer observes, that wolves and other creatures have been observed to walk around fires (particularly campfires) in a circular path. If the screenplay's conceit is that the war is a fire that Chris Kyle figuratively circles because he has an irresistible desire to go on the attack, then the wolf metaphor is apt.
But another way to understand the scene is by applying the moth-to-a-flame metaphor that Dan Sheppard suggests in a comment above. Here, the moth's fascination with light (a flame) leads it to circle the light source and eventually (if it is unlucky) to blunder to close to the flame and catch fire. Metaphorically this would describe Chris's fascination with war—a fascination that puts him at physical risk and prevents him from turning his attention away from its dazzling brilliance.
The more familiar expression of the moth-and-flame metaphor is "like a moth to a flame," which a Google Books search finds as early as Mary Tincker, Aurora, serialized in Lippincott's Monthly Magazine (May 1885):
"I am glad they have got the petroleum out of Pennsylvania," Miss Fisher remarked. "And perhaps we may be able to use up all this fuel before it has tome to catch fire."
"In that case," retorted the other, "the central fire would go out, the machine would stop, centrifugal power would cease, and we should fly to the sun like a moth to a flame."
But the circling moth variant has been around for a while, too. Google Books and Elephind searches turn up two matches for it from the early twentieth century.
From Harry Greene, The Lash of Circumstance, serialized in the Record Journal of Douglas [Colorado] (February 2, 1912):
In my uncle's large front room there was a safe that he had picked up somewhere in the dark ages of his youth, doubtless upon some foreclosure; and it was on of the simplicities of his complex mind that he should cling to it with the faith which a child reposes in the toy savings bank in which he had deposited his treasure. It was a dogged enough contraption, and around it revolve the principal characters and events of this history as a moth circles a flame.
And from Hardy Hoover, "The Man who Understood Women," in The Michigan Chimes (December 1921):
Having constructed the conventional triangle, we can now proceed with the story. The astute reader has already guessed that the two men are to fall in love with the one woman. Proceeding logically, we have the spectacle of two moths revolving enthusiastically and recklessly about one flame. This produces a mixed metaphor which the reader will instantly appreciate. We now have two well-motivated moths circling a flame, all within an isosceles triangle. With a metaphor in one hand and a triangle in the other, let us proceed.
And here is a more fatal account of a figurative moth and the object of its desire, in Jerome Gilson, The Soviet Image of Utopia (1975):
The utopianism of the Brezhnev era has not reached—and may never reach—the level of the earlier [Khrushchev] period, but it does demonstrate a central and absolutely decisive point: there can be no Soviet system of government, with its legitimate one-party monopoly of power, without a continuing, viable mission of building communism as its justification.
Like a moth circling a flame, the Soviet regime is drawn by its very nature closer to its all-consuming end. It cannot fly away, for that would deny its nature, nor can it stay at a safe distance forever. The goal is part of the Party's essence, yet the goal, when reached, means the Party's disappearance. The conservatism of the current regime and even its suppression of dissident voices in Soviet society are only means to the self-destructive end. As long as there is a one-party state in the Soviet Union, and as long as Marxism-Leninism is its legitimizing ideology, there will be a utopian goal known as communist society to justify it and all its acts.
The chief problem for the moth-circling-a-flame hypothesis is that Taya Kyle uses the plural form flames rather than the singular flame. But whether that detail is disqualifying depends on your sense of how linguistically precise and well informed one imagines (a) the character Taya Kyle and (b) the screenwriter Jason Hall to be. I wouldn't hold either one to an especially high standard.