Plot hole: When M-5 destroys the ore freighter, Dr. Daystrom says, "Fortunately, it was only a robot ship." But Daystrom wasn't on the bridge yet when Spock announced that fact. He didn't have any way of knowing that the freighter was unmanned. (00:23:00 - 00:25:05)Jean G
Plot hole: Scotty is receiving images of the launch, supposedly from an orbiting satellite. But many of the camera views are obviously from ground level, and couldn't possibly have been taken from orbit. It's not a rebroadcast from a ground camera, either. The implication is clearly that Scotty is is picking up a live satellite feed. (00:34:30)Jean G
Plot hole: Spock claims that history is unchanged at the end of the episode according to the library tapes - how would he know? If history changed, the tapes would change too. And unlike in "City on the Edge of Forever", there's no Guardian around to keep people from being influenced by an altered timeline.
Plot hole: The crew of the Galileo are stranded because there is a "dampening field" preventing the shuttle craft and its equipment from working. But the device they use to short out the "companion" works.