It is usual when a soldier is buried in a formal grave for a volley of shots to be fired over the grave by a squad of his colleagues. The fact that no soldier fired over this particular grave is either to indicate that he died in wartime and was buried where he fell, or maybe he wasn't a soldier but he was not recognised as a hero when he died and therefore received no honours from his country when he died.
Full context available.
This is from The Burial of Sir John Moore after Corunna
Not a drum was heard, not a funeral note,
As his corse to the rampart we hurried;
Not a soldier discharged his farewell shot
O'er the grave where our hero we buried.
General Sir John Moore was indeed a soldier.
He was mortally wounded at Corunna in 1809.
I don't know why he was buried quietly at dead of night, but it may have been related to the fact that the British were fighting a rearguard action to defend their evacuation from Corunna.
Charles Wolfe was the curate of Donaghmore, a small rural parish not far from here. The poem was first published in the Newry Telegraph.