Deliberate mistake: When Tarzan fights with the leopard, it scratches his chest. However after they fall into the pit and Tarzan comes out carrying the dead leopard, the scratches on his chest have amazingly healed. [It is a long-standing Disney tradition that no evidence of violence is shown on screen. Still a mistake, but there's why].
Continuity mistake: In the scene where Jane is telling her father about Tarzan, and Clayton calls it a "girlish fantasy", Tarzan jumps into the camp. Throughout the entire scene, Jane's hair was slightly tied back, but when Tarzan comes very close to her and says "Jane", it's down. The next shot it's slightly up again.
Continuity mistake: As young Tarzan fixes his spear, he's sitting on a moss covered branch, and just behind him on his left is another branch growing from the one he's sitting on. After the shot of the yellow fruit, it returns to Tarzan. Assuming he hasn't moved from his spot, the branch that was behind him should still be there, but isn't.
Continuity mistake: When Tarzan is climbing the tree as "Son of Man" starts, the continuity of the bark, vines and moss on the tree doesn't match between shots. Example: Just before he begins to climb, there's no bark on the tree. When he reaches up, there's bark. After he slips, he stops in a barkless area, but it's the same area where there should have been vines and moss as there was previously. After Kerchak passes him, there's more bark on the tree than there was in the previous shot, and so on.