In wide shots of the Golden Gate Bridge with downtown San Francisco in the background, the skyline-dominating Salesforce Tower can clearly be seen. This building did not break ground until 2013, over twenty years after this movie takes place.
When testing Bee's new radio/tape deck, Charlie briefly plays a tape of "Never Gonna Give You Up"; although the song is from 1987, the year of the film's setting, it did not release in the United States until early 1988.
While Charlie's mom is driving Bumblebee, she freaks out when Charlie pulls up next to the driver's window in her moped and says that she was afraid she was being carjacked. The term carjacking was not used in 1987. Per Wikipedia: The word is a portmanteau of car and hijacking. The term was coined by EJ Mitchell, an editor with The Detroit News. The News first used the term in a 1991 report on the murder of Ruth Wahl, a 22-year-old Detroit drugstore cashier who was killed when she would not surrender her Suzuki Sidekick.
In one sequence, Bumblebee is seen crushing a can of Tab Cola with his fingers. The can however is of modern vintage, as shown by the can's shape (with the tapers, as opposed to the more purely cylindrical can of 1987) and the calorie count mark on the label (which was added sometime after 2010)
In the scene where Charlie and Memo go to the big cliff-side party, at one point Memo says the phrase "What's your damage?" to one of the bully girls. This phrase was popularized by the movie Heathers, which came out in 1988, the year after Bumblebee is set.
During the mid-credits scene, Bumblebee meets with Optimus Prime and they witness other Autobots arriving on Earth. In the original film, none of the other Autobots arrive until 2007, two decades after the events of this film.
At the very end, as Bumblebee is seen from the front, driving across the Golden Gate Bridge, his headlights are on. When seen from the back, the taillights are not on - either they're on or off, but not both.
In the original movie continuity, Bumblebee is said to have had his voice damaged in a battle with Megatron and Ratchet had been trying to fix it. In this movie his voice box is ripped out by Blitzwing. However, the 2007 movie itself never establishes that it was Megatron who took Bumblebee's voice, only the tie-in comics which already have been contradicted by the other movies in many ways. As this is the only movie that in fact shows how Bumblebee lost his voice, it should be taken as canon over any previous statement, even if it is a retcon.