The Apollo 11 launch site is referred to as Cape Canaveral. While this would have been natural for Agent J, it would have been unnatural for young Agent K as the Cape was re-named Cape Kennedy after President Kennedy's assassination in 1963. The name only reverted back to Cape Canaveral in 1973, way after Apollo 11 was launched.
The slidewire astronaut escape baskets were not installed until after the Apollo era, and were used for the Space Shuttle. Apollo astronauts were expected to descend in a fast elevator, then retreat to a bunker under the pad in case of emergency.
In 1969 every baseball cap had a green underbill. In the film you can see a 1969 Mets pitcher using a gray underbill cap. Gray underbill caps aren't introduced until 1994. In 2007 this color was dropped in favor of the current black.
Also during the Andy Warhol party there are lights seen on the stage curtains, commonly called a Star Cloth or Curtain today. This technology as displayed in the film did not exist in 1969, in fact any sort of mass production of very low powered Red LEDs only started in 1968 and prior to that most LEDs cost in the vicinity of $200 each!. White LEDs did not exist then.
When Agent J gets pulled over by the two Police Officers while driving the stolen Cadillac, the light powder blue shirts were not worn in the NYPD until 1973. The correct uniform would have been dark navy blue shirts without NYPD patches on the sleeves, which were not designed as the department emblem until 1973 as well.
The location of the 1969 party scene with Andy Warhol was clearly meant to represent his "Silver Factory," so called because the walls were covered with aluminum foil. The Silver Factory closed in 1968.
Papua New Guinea gained independence in 1975. The flag of Papua New Guinea was adopted on July 1, 1971. In 1969, the current flag of Papua New Guinea did not exist, however, it can be seen prominently in several shots during the speech before the launch of Apollo 11.
The Spanish flag flapping in the wind during the Cape Canaveral launch scene is the current Spanish flag. In 1969, Spain - under the Francoist regime, had a different flag (bearing a coat of arms with a black eagle).
When J, K and Griffin enter MIB headquarters at the Brooklyn-Battery Tunnel ventilation building to fly to Cape Canaveral, the tilting shot prominently features three medallions over the entrance doors. Those cast aluminum reliefs, designed by Paul Manship, would not have been over the building entrance in 1969. They would have been mounted on the New York Coliseum convention center in midtown Manhattan. The reliefs were removed from the Coliseum in 1999 as it was being prepared for demolition and only later mounted on the ventilation building.
During the Andy Warhol party scene the stage curtains are seen supported by an aluminum truss structure. In 1969 this type of system did not exist and any truss from that era would have been made from steel.
Throughout the movie, especially in the beginning, Boris the Animal insists his name is "just Boris!" Yet when Agent J visits Jeffrey Price's shop and sees the guestbook, Boris signed his name as "Boris the Animal."
When the 1969 cops have J pulled over, before pressing the button on the neurolyzer, he is not wearing his glasses, but in the next shot, he has them on, despite the fact that he didn't have a free hand to put them on.
At the stadium, when Griff is showing J and K the future game, right after he says the line "A miracle is what seems impossible...", in the shot from the back his hands are on their backs but on the shot from the front he is holding their hands by the elbows
When J and K first encounter Boris on the rooftop of Wu's restaurant, Boris forces them off the roof, hanging on a door. When J and K hit the ground, there is a brief shot showing them falling in opposite directions. However, the very next shot shows them on the ground, side by side.
When on the moon in the beginning, the Apollo lunar module is shown with the upper ascension module. This module was launched off of the moon to return the astronauts to the orbiting module to return to Earth.
After escaping from Lunar Max prison, Boris the Animal is standing on the lunar surface and can be heard saying "Let's rewrite history, shall we, K?" The moon has no atmosphere, so there is no way of transmitting sound.
Young Agent K is instructed to place the pendant on the Apollo escape rocket to get it into space to protect the Earth. The escape rocket never goes into space. Once the Saturn V is high enough for the capsule to parachute safely in an emergency, the escape rocket is jettisoned.
J jumps from the roof of the Chrysler Building, which is 925 feet high. He falls for more than 30 seconds before activating the time jump device. Given the height of the building and the acceleration of gravity (32ft/s2), it should take less than 8 seconds to reach the ground. In the amount of time J spends falling (more than a minute), he could in reality fall from a building 75 times as tall as the Chrysler building.
At Cape Canveral when J, K, and Griffin meet the Colonel (and all scenes after that with the Colonel), the Colonel is wearing a brown belt with a brown holster. In 1956 the US Army officially changed the color for belts, holsters, and shoes from Cafe Brown to Black. In 1969 the Colonel should have a black holster, and belt like the M.P.s are wearing.
The lunar module seen outside of the prison is presumably The Eagle of Apollo 11. In that case, the flag would not be standing (the astronauts reported seeing it flop over when they lifted off from the moon, unless, of course, someone replanted it).
The motorcycle ridden by Boris the Animal has an low-profile rear wheel and tire which were not made in 1969. However, given that the motorcycle burns with a blue-green flame after it crashes, it is probably reasonable to assume it is at least partly of extraterrestrial manufacture.
In the first Men in Black, when K is deleting J's history; He deletes his birthday which was in 1975. The third film is set in 1969 and when K meets a young J, he is about 4 or 5. This would mean he was born no later than 1965, not 1975.
J tells Young K that 39 years into the future Boris escapes from prison to time travel back and alter the future. That would put the modern day events of the film in the year 2008. Yet the display on J's time jumper shows he came from the year 2012.