Continuity mistake: When Bowman attempts to rescue Poole, he enters "C-pod" which, from HAL's vantage point, is the right-most pod. But when we see a close-up of the pod's door closing, we see it is located on the center pod platform, meaning this is B-pod, not C-pod - yet B-pod is the one Poole was using and it is tumbling away in space at this time. We can also see, just barely, that all three pods are present in this same shot.
Continuity mistake: During Heywood's meeting with Elena and Dr Smyslov, he is introduced to their colleagues, including Dr Kalinan who is the women in the olive dress/suit. When they shake hands her coat is folded neatly over her seat, but before she sits down again with the camera now facing Heywood, the coat disappears. In an earlier shot prior to the introduction, the majority of Kalinan's coat is off the seat and when she stood the first time it probably fell, suggesting the handshake sequence was re-filmed after the actual meeting.
Factual error: The scene at the Tycho excavation site where the uncovered monolith emits a radio signal after being illuminated by the rising sun incorrectly shows the sun directly overhead. This would be impossible as the crater Tycho is in the moon's southern hemisphere and the sun would never rise that high during the lunar summer.