When Professor Robinson is contemplating turning off the main shut off switch he worries that it will, "turn off the oxygen and destroy us all." This makes no sense as the planet consists solely of plant life which consumes carbon dioxide so there would be no need for an oxygen supply. The plants emit oxygen so the supply of that would not be affected.
The vest part of the carrot suit has 2 strips of duct tape around the waist to hold it flat,
During the cold open, it's mentioned twice that there is only vegetation on the planet and no animal life, yet when Smith lands, birds can be heard.
At 8:13 you can see the glove and wristwatch up the sleeve of the carrot's left arm.