When prof. Perlman is showing Oliver the colour slides of Greek statues sent from Berlin, the 4th one is the 'dancing satyr' which was discovered in Sicily in 1998 (the statue is displayed in a museum specially built for it in Mazara del Vallo in Sicily), while the film is set in 1983.
Elio's wears a watch that appears to be the ubiquitous Casio F91-W, however this watch did not enter production until 1991 and despite looking retro doesn't quite match the setting of 1983.
Continuity error after Elio grabs Oliver's crotch. As shot from the front, Oliver removes Elio's hand with his left hand. Cut to the immediate continuing shot from the rear but Oliver now has Elio's hand in his right hand as he sets Elio's hand back on Elio's knee.
In the final shot of Elio at the fireplace, a housefly can be seen crawling on his shirt. Flies don't live long enough for it to be in the house that time of year, when snow has fallen (and during Hanukkah).
In the Hanukkah coda at the end of the film when Elio enters the kitchen and exclaims "Oooh, latkes!", Elio reaches down and takes two latkes from the stovetop with his left hand and exits the kitchen with the latkes in hand. Cut to Elio in the hallway outside the kitchen, no latkes in hand.
When Elio and Oliver are at the car preparing to head off to Lake Garda with Sam, Elio has his right foot up on the open back door panel during their conversation. However, when the shot changes to Oliver's point of view in the front seat, Elio clearly has his left foot up on the door frame. Then back outside and it's right foot again.
In the final scenes the Hanukkah hanukkiah is lit incorrectly for the 7th night. It is customary to place the candles on the right side of the hanukkiah. The missing 8th candle should be on the left side.
Elio wears an anachronistic Fido Dido T-shirt a couple times in the movie, including the beautiful scene on the sofa with his father near the end of the film. Although the film takes place in 1983, the Fido Dido character wasn't created until 1985. And in fact, if you pause/freeze-frame playback during that scene on the sofa, you can clearly see on the front of Elio's T-shirt the copyright year 1985 in small lettering (showing as "© 1985 Fido Dido") on the lower left side.
In one scene just outside the Perlman's house, it is possible to see three cans of Illy coffee; in 1983, the design of the cans was different: shorter and larger. On top of that, the Illy logo displayed on the cans on display was created by James Rosenquist only in 1996.
The position of Elio's hands change in each edit during the first dinner party.
As soon as Oliver's train has left the station of Clusone, Elio calls his mother on the phone, asking her to come and pick him up. The train station of Clusone was a bus only station by 1983, as the railway line of Valle Seriana (in the province of Bergamo) closed down in 1967 to be substituted by a bus line.
During the dance party, when the DJ changes the song to "Love My Way," the girls in the foreground react in excitement before it's possible to even register what the new song is.
This film gets the cars right except for the Alfa 90 released in 1984 whilst it's still 1983 in the film.
During the foot massage scene, Elio's arm position changes drastically between cuts.
The Persol sunglasses Oliver wears throughout would have had a 4-barrel "Meflecto" temple in 1983; only after Luxottica bought the company in the 2000s did they reduce the 649 to 2 barrels, which is seen in the film.
In the breakfast after the night Elio swam with Marzia, Elio was making a crepe with Nutella, the crepe had been wrapped, but next shot Elio was making the crepe with Nutella again, and next shot he took the wrapped crepe again...
The book that Elio gave to Marzia is the collection of poems "Parole" [Words] by Antonia Pozzi, curated by Alessandra Cenni and Onorina Dino and published for the first time by Garzanti in its series Poesia in 1989 (while the movie is set in 1983).
Throughout the entire movie Oliver wears a Star of David necklace. In the final scene where we see Oliver, saying goodbye to Elio at the train station, he is not clearly seen wearing the Star of David necklace. No explanation is given for why he would take it off or not be wearing it. Though it's very very hard to clasp a glimpse of it due to the way the shirt is buttoned up, there are a few split seconds where you can see the necklace is definitely on his neck during that scene.
On the morning after, Oliver is laying with his arm around Elio's shoulder and Oliver's bike wound almost looks healed, next scene after the swim when Oliver walks up to Elio the wound looks scabbed over again.
Elio states at the monument for the Battle of Piave that 170,000 died there. In fact, while there may have been approximately 170,000 casualties (dead,wounded,captured ) , the total number of dead was probably closer to 20,000.
When Elio grabs Oliver's crotch, Elio's legs are outstretched but when Oliver moves Elio's hand back, he places it on Elio's knee, showing Elio's legs pulled up.
When Elio is seated in the garden, just before being shown a fish by Anchise, there is a Sony HF cassette tape on the table which dates from around 1988. A correct one for the era could be the Sony CHF with white & orange labels on the shell.