Cue Tim Krabbe's Chess Record! He lists immobility records for each piece, but since you specify never moving, here are the fitting games. Alas, I know not if any players involved were GMs at the time, but records are records!
I pulled the move lists from Krabbe's somewhat secretive counterpart Palview viewing page of these games.
Ng8 sat immobile for 85 moves.
[Title "Galanov-Kosanski, Budapest 1994"]
[FEN ""]
1. e4 g6 2. d4 Bg7 3. Nc3 d6 4. Be3 c6 5. Qd2 b5 6. O-O-O Nd7 7. e5 d5 8. h4 Nb6 9. h5 Nc4 10. Bxc4 bxc4 11. Na4 Bf5 12. Ne2 Rb8 13. f3 gxh5 14. Nf4 e6 15. Nxh5 Bf8 16. a3 Qc7 17. g4 Bg6 18. Nf4 Qb7 19. Qc3 h6 20. Rh2 Qb5 21. Nc5 Bxc5 22. dxc5 Kd7 23. Rdh1 a5 24. Bd4 a4 25. Qe3 Rf8 26. Ne2 Rb8 27. Bc3 Re8 28. f4 Be4 29. Rd1 Kc8 30. Bd4 Qa6 31. Ng3 Bg6 32. f5 Bh7 33. Qf4 Qb7 34. Rdh1 Qd7 35. f6 Bg6 36. Nh5 Qb7 37. Qd2 Qb5 38. Qc3 Kd7 39. Rg1 Rb8 40. Nf4 Rh7 41. Rh4 Be4 42. Nh5 Qa6 43. Ng3 Bg6 44. Rgh1 Rb5 45. g5 Rb8 46. gxh6 Qc8 47. Nf1 Qf8 48. Qe3 Re8 49. Qf4 Be4 50. R1h2 Rb8 51. Qe3 Bg6 52. Nd2 Rb5 53. Nf3 Bf5 54. Bc3 Rb8 55. Bd2 Rb7 56. Qg1 Bg6 57. Qd4 Ke8 58. Be3 Rb5 59. Qc3 Kd7 60. Nd4 Rb8 61. Qa5 Ra8 62. Qb6 Qc8 63. Nf3 Rb8 64. Qa5 Rb5 65. Qa7+ Rb7 66. Qxa4 Qb8 67. Bd4 Qf8 68. Qa6 Rc7 69. Be3 Qb8 70. Qa5 Ra7 71. Qc3 Qf8 72. Qd4 Ke8 73. Qd2 Kd7 74. Rxc4 Qa8 75. Rch4 Ra4 76. Rxa4 Qxa4 77. Qb4 Qa6 78. b3 Qf1+ 79. Kb2 Qxf3 80. Qb7+ Kd8 81. Qb8+ Kd7 82. Qd6+ Ke8 83. Qxc6+ Kd8 84. Qb6+ Ke8 85. c6
Bc8 sat immobile for 81 moves.
[Title "Levitt-Lev, Tel Aviv 1989"]
[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. e3 c5 5. Nge2 cxd4 6. exd4 d5 7. c5 Ne4 8. Bd2 Nxd2 9. Qxd2 O-O 10. a3 Ba5 11. g3 Nc6 12. b4 Bc7 13. Bg2 b6 14. f4 Rb8 15. Qd3 Qe7 16. b5 Nd8 17. c6 f5 18. Qe3 Qf6 19. O-O Nf7 20. Kh1 Nd6 21. Qd3 h6 22. Ng1 a6 23. a4 Ra8 24. Nf3 Ne4 25. Qe3 a5 26. Ne5 Bd6 27. Bxe4 fxe4 28. Ra2 Ra7 29. Raf2 g6 30. Qe2 Rg7 31. Nd1 Qe7 32. Ne3 Bc7 33. Qb2 Kh7 34. Kg2 Kg8 35. Qc3 Qd8 36. Qd2 Qe7 37. Nc2 Bd6 38. Qe3 Bc7 39. h3 Bd8 40. Qa3 Qc7 41. h4 Be7 42. Qe3 Qd8 43. Kh3 Kh7 44. Qe2 Qe8 45. Ne3 Bd8 46. Qa2 Qe7 47. Qb2 Bc7 48. Rh2 Bd8 49. Kg2 Qe8 50. Kf2 Bc7 51. Ke1 Qd8 52. N3g4 Rf5 53. Rfh1 Qe7 54. Rf1 Qe8 55. Qa3 Qd8 56. Ne3 Bd6 57. Qb2 Rf8 58. Kd1 Qe8 59. Qe2 Qd8 60. Rhf2 Qe8 61. Qg4 Bc7 62. Qh3 Bd6 63. h5 gxh5 64. g4 Bxe5 65. fxe5 Rxf2 66. Rxf2 Qe7 67. Qxh5 Qa3 68. Nc2 Qd3+ 69. Kc1 e3 70. Rf6 Qd2+ 71. Kb2 e2 72. g5 Qxg5 73. Qxe2 h5 74. Qd3+ Kh8 75. Qf3 Rg8 76. Rf7 h4 77. Ne3 Rg7 78. Rf8+ Rg8 79. Rf4 Qh6 80. Qf2 Rg3 81. Rf6
Rh1 sat immobile for 105 moves. This is not a grandmaster game; nonetheless, it is a record holder. It also holds the record for the longest series of checks by one side.
[Title "Rebickova-Voracova, Girls under 16 ch, Czech Republic 1995"]
[FEN ""]
1. e4 e5 2. d4 Nc6 3. d5 Nd4 4. Be3 Bc5 5. Nf3 Qf6 6. Bg5 Qb6 7. Nc3 Qxb2 8. Na4 Nxc2+ 9. Ke2 Qxa1 10. Qxc2 d6 11. Nxc5 dxc5 12. Qxc5 Qxa2+ 13. Nd2 a5 14. Qxc7 Nf6 15. Qxe5+ Kd7 16. g3 Qa3 17. Bh3+ Kd8 18. Bxf6+ gxf6 19. Qxf6+ Qe7 20. Qxh8+ Qe8 21. Qf6+ Qe7 22. Qb6+ Qc7 23. Qb2 Qe7 24. Qg7 Bxh3 25. Qxh7 Bg4+ 26. f3 Bd7 27. Nc4 Bb5 28. Kd3 Bxc4+ 29. Kxc4 Qc7+ 30. Kd4 a4 31. Qg8+ Kd7 32. Qxa8 Qb6+ 33. Ke5 Qd6+ 34. Kf5 Qg6+ 35. Kf4 Qf6+ 36. Ke3 Qc3+ 37. Kf2 Qd2+ 38. Kf1 Qd1+ 39. Kg2 Qd2+ 40. Kh3 Qh6+ 41. Kg4 Qg6+ 42. Kh3 Qh6+ 43. Kg4 Qg6+ 44. Kh3 Qh6+ 45. Kg4 Qg6+ 46. Kf4 Qf6+ 47. Ke3 Qc3+ 48. Ke2 Qc2+ 49. Kf1 Qd1+ 50. Kg2 Qd2+ 51. Kg1 Qe3+ 52. Kg2 Qd2+ 53. Kh3 Qh6+ 54. Kg4 Qg6+ 55. Kf4 Qf6+ 56. Ke3 Qc3+ 57. Kf2 Qd2+ 58. Kg1 Qe3+ 59. Kf1 Qc1+ 60. Ke2 Qc2+ 61. Ke3 Qc3+ 62. Kf4 Qf6+ 63. Kg4 Qg6+ 64. Kh3 Qh6+ 65. Kg2 Qd2+ 66. Kf1 Qd1+ 67. Kf2 Qd2+ 68. Kg1 Qe3+ 69. Kg2 Qd2+ 70. Kh3 Qh6+ 71. Kg4 Qg6+ 72. Kf4 Qf6+ 73. Kg4 Qg6+ 74. Kh3 Qh6+ 75. Kg2 Qd2+ 76. Kg1 Qe3+ 77. Kf1 Qxf3+ 78. Ke1 Qe3+ 79. Kd1 Qd3+ 80. Kc1 Qc3+ 81. Kb1 Qb3+ 82. Ka1 Qa3+ 83. Kb1 Qb3+ 84. Kc1 Qc3+ 85. Kd1 Qd3+ 86. Ke1 Qe3+ 87. Kd1 Qd3+ 88. Kc1 Qc3+ 89. Kb1 Qb3+ 90. Kc1 Qc3+ 91. Kd1 Qd3+ 92. Ke1 Qe3+ 93. Kf1 Qf3+ 94. Kg1 Qe3+ 95. Kg2 Qxe4+ 96. Kh3 Qh7+ 97. Kg2 Qc2+ 98. Kf1 Qf5+ 99. Ke2 Qe4+ 100. Kd2 Qxd5+ 101. Kc3 Qc5+ 102. Kd3 Qd5+ 103. Ke3 Qe5+ 104. Kf2 Qf5+ 105. Kg2 Qc2+
In this game, none of the four rooks moved whatsoever!
[Title "Szabo-Ristoja, Helsinki Finland, 1975"]
[FEN ""]
1. d4 Nf6 2. c4 c5 3. d5 e5 4. Nc3 d6 5. e4 Be7 6. Bd3 Nbd7 7. Nge2 Nf8 8. Ng3 g6 9. h4 h5 10. Nf1 Ng4 11. f3 Nh6 12. g3 a6 13. Qc2 Bd7 14. a3 Qb8 15. a4 Bd8 16. a5 Bc7 17. Bg5 Ng8 18. Nd1 f6 19. Bd2 Kf7 20. Nfe3 Nh6 21. Nf2 b6 22. axb6 Bxb6 23. f4 a5 24. Be2 Kg7 25. f5 Be8 26. g4 hxg4 27. Nfxg4 Nxg4 28. Bxg4 a4 29. fxg6 Nxg6 30. Nf5+ Kg8 31. h5 Nf4 32. Bxf4 exf4 33. h6 Ba5+ 34. Kf1 Kf8 35. Qf2 Qb3 36. Qxf4 Bc7 37. h7 Bf7 38. Qh6+ Ke8 39. Qxf6 Qd3+ 40. Kg2 Qxe4+ 41. Bf3
Here is a honorable mention regarding kings though. Neither wiggled once in 47 full moves!
[Title " Thorhallsson T.-Johannesson, Hafnarborg Iceland, 2003"]
[FEN ""]
1. e4 e6 2. d4 d5 3. Nc3 Bb4 4. Nge2 Nc6 5. e5 Nge7 6. h4 h6 7. h5 f6 8. exf6 gxf6 9. a3 Ba5 10. b4 Bb6 11. Na4 e5 12. Bb2 e4 13. Bc1 Bg4 14. c3 Qd7 15. Bf4 Nd8 16. Qc2 Ne6 17. Bd2 Qc6 18. Qb3 Nf5 19. Rc1 Nd6 20. Rh4 Bxe2 21. Bxe2 a6 22. Rg4 Ng5 23. Nc5 Ba7 24. Rf4 Rf8 25. a4 Bxc5 26. bxc5 Nc8 27. Rb1 Rb8 28. c4 Ne7 29. cxd5 Nxd5 30. Bxa6 Qxa6 31. Qxd5 Rd8 32. Qxb7 Qxa4 33. Be3 f5 34. Qxc7 Ne6 35. Qe5 Qa2 36. Rc1 Qd5 37. Rh4 Qxe5 38. dxe5 f4 39. Bd2 Rd5 40. c6 Rf7 41. c7 Nxc7 42. Bxf4 Ne6 43. Bxh6 Rxe5 44. Be3 Rff5 45. h6 Rh5 46. Rxh5 Rxh5 47. Rc4