Deliberate mistake: When "Legs" lowers the Pez duck named Ducky down through the removed lightbulb hole, to ring the doorbell, the lightbulb housing is bigger than the hole but they bring it up through the hole to remove it.
Continuity mistake: When Buzz is in Sid's sister's room, Woody comes up to it disguised in the Christmas lights. He then leaves them beside the door before he goes in. When they leave the room, the lights are gone. Later the lights return when Woody goes to get them, but there are fewer of them.
Other mistake: It's been mentioned that Sid sets his clock for 8:25 and it goes off at 7:00. Here's some more errors for Sid's crazy clock physics. When Woody is saying "Psst" to Buzz from inside the blue plastic basket, the clock on the wall says 6:25, yet the alarm clock next to Buzz says 6:55. It goes off 5 minutes later and after Sid takes Buzz out of the room, the clock on the wall has changed to 3:10. When Woody is talking to Sid's toys a minute later, the clock now reads 10:00. The maximum amount of time that passed between the already wrong 3:10 and the 10:00 is 5 minutes. You can tell because Sid wanted to launch Buzz strapped to the rocket ASAP and Woody wanted to rescue Buzz from his current strapped-to-a-rocket state very fast.
Continuity mistake: When the army soldiers are yelling into the baby monitor that the kids are heading up to Andy's room, the baby monitor is right in front of Woody on the floor. Yet, when the camera pans out to show all the toys scrambling for their places the monitor is gone, and Hamm is right in front of Woody. Then, when the camera zooms back in to show Woody, the baby monitor has reappeared.
Continuity mistake: When Andy's mom suggests to Andy they go out to eat at Pizza Planet and that Andy can only bring one toy, Woody picks up the ball (the one that can predict things) which was next to him and asks if Andy will pick him. If you see in the earlier shots the ball wasn't there.
Continuity mistake: When Woody and Buzz are at the gas station and the big truck pulls in, Woody falls limp to the ground. The truck pulls over him, and stops with its tires an inch from his head. In fact, the brim of Woody's hat sticks up into the space between the two tires. In the next shot when Woody scrambles out from beneath the tires, his head is directly under the tire to our right.
Continuity mistake: When Andy goes to his room to get Molly, he lowers the side of her crib in order to get her out, picks her up, and walks out of the room. In the next shot of his room, the crib side is back up.