When Josh wakes up as an adult, his underwear seems the right size, but if he wore them the night before, they should surely have been too small for him.
The morning after Josh gets "big", he tries to convince Carol that he is really her son. He tells her that his birthday is November 3. Later, there is a milk carton with a "missing kid" advertisement on the side about him, which says that his birthday is January 20.
When Josh first meets Billy after turning older, a dodgeball on the floor by him appears and disappears between shots.
When Josh meets Billy for the first time in the ballroom, the football helmets change position on the table beside him throughout the shots.
(at around 37 mins) After Josh is "shot" by the toy laser gun at FAO Schwarz, he falls down to the floor quivering. The black laser power box is hanging down on the right side of his chest, with him on his back. When Robert Loggia asks, "You work for me, don't you?" it's hanging down on the left side of his chest.
When Josh calls Carol, even in 1988 police (or at that stage, FBI) would have had some sort of track-and-trace on the phone line, and the movie would have ended with him being arrested at MacMillian's Toys. This could have been easily addressed by having him call from a payphone.
In the closing credits where acknowledgments are given to entities who assisted in the movie, the entry for "FAO Schwarz" is misspelled "FAO Schwartz."
When Josh is filling out the job application, he doesn't know his social security number. Billy gives him 6 numbers to write down (his locker combination). Then at the interview, the interviewer tells him, "There's a couple of numbers missing on your social security." He then says, "Oh, 12." The interviewer then mutters "0-1-2" and writes them in, presumably at the end. This would give the number the correct number of digits (nine), but with the numbers Billy gave him, the number would start with "321", which is for Illinois residents. Josh is obviously from New York. Additionally, the personnel department should be able to run the number and find that it either doesn't exist or is not assigned to Josh.
When Josh replies to the computer game's question "What do you want to melt him with?", he hits the same key over and over.
In Times Square, the movie Wall Street is playing, but it was released in December 1987. The milk carton with Josh's picture on it shows that he went missing in September 1987.
When Carol is freaking out when big Josh walks in, she is speaking and screaming but her mouth isn't moving.
At the MacMillan party, the upright bass player at one point stops playing and says something to another man, then carries on playing. You can still hear the bass even though he stopped playing.
When Josh and Paul are walking to the court, Paul says a line but his mouth clearly says something else.
When Josh and the St. James Hotel man are talking about the $10 deposit for the sheets, the man's voice continues into the next shot without his mouth moving.
Billy's mouth as he says, "...all the way down into her flowers."
The fair at the beginning of the movie was set in New Jersey (the George Washington Bridge was visible). Josh and Billy filed an information request with the New York City Department of Consumer Affairs, which wouldn't cover New Jersey. They would have needed to go to the equivalent office for Bergen County.
When Josh is on his job interview, the interviewer tells him that he "seemed to have forgotten a few numbers on his social security number". He apparently does not know it and even makes up the last few numbers, which the interviewer writes down. If it was indeed incorrect, then the payroll department of McMillian Toys would have not been able to process any taxes from his paycheck, thereby he would have not received one when they were handed out.
In the scene in which Josh and Susan are jumping on the trampoline, a boom microphone is reflected in the window.
When Josh and Susan are standing up through the sunroof of the limo after leaving the party, the shadow of a boom microphone is visible several times on Josh as the limo passes under streetlights.
Before Josh and MacMillan play the Walking Piano at FAO Schwarz, Josh throws a bag down on the ground. When they leave he leaves it there and doesn't take it with him.
When Billy's mother is complaining at the dinner table, he goes to remove a roast from the oven. When he takes it out using a potholder, he also grabs it with his other hand which would result in him being badly burned.