"Dulce et Decorum Est" is a poem full of irony, not least in the title, where the Roman poet Horace's famous observation that it is a sweet and honorable thing to die for one's country is repeated, only to be dismissed as an "old lie." The "haunting flares" on which the soldiers turn their backs contribute to this atmosphere of uncertainty mixed with anger. The phrase is almost an oxymoron. A flare is a streak of fire cutting through the sky, only of any use if it is bright and vivid. The word "haunting," by contrast, suggests something pale and indistinct, a gray ghost hovering in the background of the scene.

The primary purpose of a flare, however, is to signal distress. Even as the soldiers turn their backs on the flares, these flares symbolize the acute pain and suffering of trench warfare. They cannot entirely turn their backs, for the flares they cannot see with their eyes haunt them in their thoughts. Vivid, fiery explosions, which look like the flares, tear off the limbs of a soldier. The pain and indignity of those lost limbs haunt the soldier forever. In this way, the two halves of an uneasy phrase add both image and symbol to the poem, reinforcing the bitter contradictions of war that it exposes.

To most people, the sight of flares lighting up the night sky might be rather an exciting experience. At the very least, it would be quite a spectacle. But this is not how the soldiers in "Dulce et Decorum Est" feel. The flares provide the bleak, haunting backdrop against which the exhausted men trudge their weary way back to the trenches for some brief, fitful rest.

In war, there are so many contradictions, and the "haunting flares" are a prime illustration of this. The bright light of the flares is a source of illumination, but not in a good way. It illuminates a scene of unspeakable horror and carnage, of broken men and bodies littering a muddy, blood-strewn battlefield. At least in darkness, the full horrors of war can be hidden to some extent. But the flares shine a revealing, probing light on the slaughter down below. They haunt the men because it is not possible to escape from them and the seemingly endless war they've come to represent. The soldiers' "distant rest" is distant indeed. They will never truly be at rest so long as the war continues, so long as the haunting flares shine their lurid light upon the men forced to participate in such grinding bloodshed.

Early in the poem, the poet works to achieve a particular mood, and the phrase "haunting flares" contributes to this.  To begin the poem, the narrator states,

Bent double, like old beggars under sacks,
Knock-kneed, coughing like hags, we cursed through sludge,
Till on the haunting flares we turned our backs,
And towards our distant rest began to trudge. (lines 1-4)

These soldiers are marching, laden with gear and so exhausted that they are hunched over, walking like old women.  They march through mud and muck, with farther to go until they are finally out of immediate danger and can actually stop to rest.  The mood is bleak, dismal; these are young men of 18, 19, 20 years old, and yet they can only "trudge" with knees knocking, gagging and hacking like a sick old woman.  The flares, then, are described as "haunting," but what typically haunts?  Ghosts, sad memories: nothing bright or good or hopeful.  The flares are haunting for the men, perhaps because  they remind them that they will face the same fate tomorrow.  The realization that they must march, pressing forward no matter what, for yet another day and another and another would surely be haunting.  The word choice of "haunting" contributes to the bleak mood as well as gives us a glimpse into the mind of a soldier in this position.  The "haunting flares" seem to symbolize the haunting knowledge that this dismal reality will be the soldiers' own for a long time.

The "haunting flares" you refer to in your question comes in the first stanza of this excellent war time poem and it is part of the overall description that helps create a slightly surreal theme of the horrors of war and the impact of war on the soldiers involved. Whenever you are trying to work out the meaning or the contribution of such phrases or lines, it is always important to look at them in context to see if that can help in your understanding of them. Let us do that now:

Bent double, like old beggars under sacks,

Knock-kneed, coughing like hags, we cursed through sludge,

Till on the haunting flares we turned our backs

And towards our distant rest began to trudge.

Note how the overall picture is one of severe exhaustion and illness. The "haunting flares" in this picture, especially through the use of the word "haunting," seem to add a definite flavour of the other-wordly, almost making the scene ghost-like as the soldiers, in their crippled, exhausted state, try to focus on heading towards the "distant rest" that awaits them.