Early on in the film when Bob rescues the cat, it's 2 completely different cats. The cat on the roof, and the cat the little girl is holding are different.
In the very beginning of the movie, Gillian tells Farren to put on jeans, and dresses Nora in pink. While getting in the minivan, Farren is wearing a skirt and Nora is wearing white.
Ian says he tracked Bob by sneaking a tracking device into his pocket when he hugged him goodbye. However, before going to track down the bad guys Bob completely changes his outfit.
When Bob is talking with Ian, the positions of the books on the shelf behind them keep changing.
When Bob Ho jumps on the planter to save the kitten, the planter is by itself. In other shots it is next to another planter.
At the beginning of the movie Bob Ho's alarm clock goes off at 6:30AM yet it's bright daylight outside. It's late October just before Halloween and the Sun doesn't even rise till about 7:30AM that time of year.
"Ho adjusts the family's satellite dish with an electronic label-maker."
The device he was using looked like a label maker because it was likely made from one. A label maker has a keyboard on it meaning with some modification it could be made to serve as an interfacing tool (which could be used to repair computer based systems like dish arrays). Building from an ordinary everyday item has the added bonus of discretion since to anyone else it would appear to be a simple inconspicuous tool or appliance.
When Larry is first introduced, he speaks perfect English, but after it is revealed that he is a Russian spy, he speaks English with a Russian accent for the rest of the movie. However, it's not inconceivable that he would drop the act once he's been found out.
Bob uses a laser to cut out a section of the grate. Although the laser moves in a circular motion, the bars are cut in a straight line.
The song that plays when the movie opens is Secret Agent Man by Johnny Rivers. But when the movie ends and the outtakes play. The song plays again. If watching with CC/subtitles, the captions will say "Secret Agent Man by Johnny Winters".
At 6:49 when Jackie Chan is surveilling the refinery, the latitude and longitudes are given as 23.5 and 26.4 resp. That puts the coordinates with the least amount of geographical knowledge possible in the northern hemisphere and eastern hemisphere. North America is in the Western Hemisphere and any longitude coordinates would always start with a negative value. For example New York is 40,-73 while LA is at 33,-117.
When testing the biological weapon, the question is asked: "What happened to the bacteria?" The scientist answers: "They live as long as they feed. Once the oil is gone, they disappear." If that were true, they could not have stored the bacteria in the first place without constantly feeding them.
When Bob mentions seeing a movie, Ian says "What movie? Octopussy? This isn't 1985." Actually, Octopussy came out in 1983.