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.
Deliberate mistake: Throughout the first part of the movie, Tarzan and all the other animals are speaking to each other in English. After Tarzan meets Jane, he has trouble communicating with her and later shows her that he communicates with the gorillas using grunts. This means that the earlier communication between Tarzan and the animals was done for the benefit of the audience. However, with this being the case, Tarzan shouldn't have been able to tell Jane his name, since the gorillas communicate with him in grunts, they would have had no way of vocalizing a name for him that could translate into a human language. At least not until Tarzan and another human could agree on what his name should translate to.
Factual error: The weight of an elephant is nowhere near that of a large cargo ship so even if Tantor was able to climb up the side of the ship the ship shouldn't have moved.
Continuity mistake: When Sabor's right leg gets caught in some rope, her tail disappears. When Sabor is seen tangled up in rope, her tail reappears.
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.