Character mistake: The Enterprise is targeted by projectiles from the planet. Spock identifies them as "plasma torpedoes", and Captain Pike reacts with surprise; "Plasma torpedoes? That's 21st century tech" and from that they all establish that the civilization they contacted is not yet capable of warp drive. Plasma torpedoes are the main weaponry of the Romulans centuries later, and Earth itself did acquire warp drive in the 21st century, so this claim and deduction don't seem to add up. (00:22:15)
Revealing mistake: When Jen is at the cash bar and keeps putting money down, it's the same shot used multiple times since it's the same serial number on the $5. And in one shot they flipped the image to make it look different (you can see the "5" is a reverse image).
Other mistake: The King's ship has no steering devices, steering wheel/pin, no steerman, and no crew. (00:26:00)
Continuity mistake: Obi-Wan Kenobi exposes Haja Estree. He asks for his help. When he hastily says "Yeah I'll do it!" Kumail Nanjiani's right hand is curled in a loose fist or fully opened, back and forth as camera angles switch. (00:08:25)