Here's the data from Dyers compendium. Looking at dates and places it appears the unit was doing garrison duty in the area above Knoxville. I've found no evidence they were in Knoxville during the seige, and since Burnsides continued to draw food supplies from the area during the seige I suspect they stayed in that area. There duties would primarily be to keep the peace and chase down the occasional raider/partisan/guerrilla. Pretty unglamorous work.
UNION TENNESSEE VOLUNTEERS
8th Regiment, Tennessee Infantry
Organized at Camp Dick Robinson and Camp Nelson, Ky., November 11, 1862, to August 11, 1363. Attached to District of Central Kentucky, Dept. of the Ohio, to June, 1863. 2nd Brigade, 4th Division, 23rd Army Corps, Army of the Ohio, to July, 1863. 2nd Brigade, 1st Division, 23rd Army Corps, to August, 1863. 2nd Brigade, 3rd Division, 23rd Army Corps, to September, 1863. 2nd Brigade, Left Wing Forces, 23rd Army Corps, to January, 1864. 1st Brigade, 3rd Division, 23rd Army Corps, to February, 1865. 1st Brigade, 3rd Division, 23rd Army Corps, Dept. of North Carolina, to June, 1865.
SERVICE.-Duty at Nicholasville, Ky., Camp Dick Robinson, Camp Nelson and Lexington, Ky., till August, 1863. Burnside's Campaign in East Tennessee August 16-October 17. At Greenville till September 19. Carter's Depot September 20-21. Jonesborough September 21. Watauga River September 25. At Bull's Gap and Jonesborough till December. About Dandridge January 16-17, 1864. Strawberry Plains January 22. Duty in East Tennessee till April