I find a Jeremiah/Jerry Fairburn listed the National Archives service records as belonging to two Va units. A Jeremiah Fairburn enlisted in I/33Va on 22 June 1861 at Harrisonburg, VA; he never reported to the unit and was never mustered in. The unit declared him a deserter on 28 Feb. 1862 and dropped him from the rolls.

A second Jeremiah/Jerry Fairburn enlisted in B/58 Va Militia at at Harrisonburg on 16 December 1861 for six months. He may have joined the unit sooner than that because one company muster roll shows him AWOL by 10 December. He was present in the unit on 31 Dec. 1861 and 23 March 1862. The 58th Va Militia, also known as the 4th Regt., 7th Brigade, Va. Militia, was called into service for a second time on 21 Oct. 1861 for six months beginning 12 Nov 1861, the date the unit was mustered in. The 58th Militia was disbanded in March 1862.

There is also a J. Fairburn listed in the NA Unfiled Slips and Papers related to Confederate soldiers. Don't know if he is connected to Jeremiah Fairburn, but he was provided transportation to Philadelphia on 11 March 1864 by the Union Provost Marshal General for Defenses North of the Potomac in accordance with the Adjutant General's General Order 64 dated 18 Feb. 1864. Union records listed J. Fairburn as a "released Rebel deserter."

Don't know if any of this helps. Was your Jeremiah Fairburn from the Harrisonburg or Rockingham County, Va., area?