I found a pdf that I had downloaded apparently in April 2007 from the Military History website re: Virginia Hospitals. I believe the author may be C. P Schulze. I have not verified the following names/places/dates, but it is a place to start. The dates represent when the facility operated.

Hospital at Martha Washington College, Abingdon, Washington Co.
General Hospital at Nottoway (now Blackstone) Co.
Burkeville General Hospital, Burkeville, Nottoway Co.
Charlottesville General Hospital, Charlottesville, Albemarle Co., 22 July 1861- 2 Aug 1865
General Hospital at Culpapper Court House, Culpepper Co., June 1861- Mar 1862
Danville General Hospital, Danville, Pittsylvania Co., 12 March 1862- 7 April 1865
Emory and Henry College Hospital, Emory, Washington Co., Dec 1862- April 1865
Farmville General Hospital, Farmville, prince Edward Co., May 1862- April 1865
Wayside Hospiral, Farmville, prince Edward Co., 1863-1865
General Hospital, Front Royal, Warren Co., 1 Sept 1861- 4 March 1862
General Hospital, Gordonsville, Orange Co., 1 June 1863- 5 May 1864
Hanover Junction, Hanover Co., 30 June 1863- 18 Oct 1864
Fairview General Hospital, Lexington, Rockbridge Co., 1863-1864
General Hospital, Liberty, Bedford Co., 2 Jan 1864- 2 March 1865
General Hospital No.1, Lynchburg, May 1861-1865
General Hospital No.2, Lynchburg, April 1862-1865
General Hospital No.3, Lynchburg, April 1862-1865
Camp Nicholls Hospital, Lynchburg, 1862-1865
Ladies Relief Hospital, Lynchburg, 1862-1865
Pratt Hospital, Lynchburg, 1862-1865
Way Hospital, Lynchburg, 1862-1865
Hospital, Mount Jackson, Shenandoah Co., June 1862
General Hospital, Orange Court House, Orange Co., 22 July 1861- 23 April 1862
Hospital of the 4th Regiment Georgia Infantry, Portsmouth, May 1861- April 1862
Hospital at Romney, Hampshire Co.
Western State Hospital, Staunton, Augusta Co.
Hospital at Warm Springs, Bath Co., 1861
Baptist Church Hospital, Williamsburg, 1861-1862
Episcopal Church Hospital, Williamsburg, 1861-1864
Methodist Church Hospital, Williamsburg, 1861-1862
Seminary Hospital, Williamsburg, 1861-1862
New Court House Hospital, Williamsburg, 1861-1862
Hospital, Winchester, Frederick Co., May 1862
General Hospital, Winchester, Frederick Co., July-Aug 1864
Lovingston Hospital, Winchester, Frederick Co., August-Sept 1862
Hospital at White Sulphur Springs, 1861

Hospitals at Richmond and Petersburg to be listed in a separate post.
Hope this info helps...so many Tennessee soldiers were treated and/or died in Virginia.