Visible crew/equipment: When the astronauts are approaching the monolith on the moon, you can see the reflection of the cameraman in the visor of the first astronaut off the ramp, in the close up of him walking around the monolith. This shot was personally filmed by Stanley Kubrick and the reflection is his own. (00:48:40)
Deliberate mistake: When Dave is in the pod arguing with Hal to let him back in, various patterns of light are projected on Dave's face presumably from the video screens that say NAV and COM and such. Light on such a screen would diffuse and not project like this. It is likely those screens had rear projection and they merely removed the screens for this effect.
Continuity mistake: When Dave begins to disconnect HAL's circuits, one angle shows him unlock the first, second, then third unit. Just as he's about to remove the key from the third lock, the angle changes and he's pulling the key from the second lock and then goes for the third again. We know this is consecutive time as HAL is pleading with Dave the entire time. (01:51:45)
Continuity mistake: When Frank Poole and Dave Bowman are discussing with HAL the possibility of HAL being in error predicting the AE-35 Unit fault, the 10x3 grid of buttons on the console in front of them is lit differently depending on whether the shot is from behind the men, or in front of them. (01:20:40 - 01:23:00)