I would believe he is buried in a cemetery close by the hospital where he died.
Elis T. Turner, Private, Company K, 41st Alabama Infantry, enlisted May 10, 1862 in Fayette County, Alabama by Col. H. Talbird for 3 years or the war, wounded and sent to Hospital Murfreesboro January 2, 1863, captured at Murfreesboro, Tenn. January 5, 1863, sent to Camp Morton, Indianapolis, Indiana prison camp, forwarded to City Point, Virginia for exchange, n.d., recorded on his companies muster roll dated August 12, 1863 as "at home in Fayette County, Ala.," on furlough, receipted for an issue of clothing May 4, 1864, and also on June 29 at Howard's Grove Hospital, Richmond, Va. having been admitted May 13 with Chronic Bronchitis, returned to duty June 30, admitted General Hospital, Petersburg, Va. July 18, 1864, gunshot wound right thigh, died August 7, 1864
M311: Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of Alabama
There is a register of patients for the General Hospital, Petersburg, Va., 1861-65, at the National Archives in Washington, DC. In record group 109, Ch. VI, vols. 265, 270, and 273. I would not know if they recorded burials.