At the beginning of the Rocky vs. Thunderlips match, the shadow of a boom mic is clearly visible on top of Thunderlip's head during the overhead shot of them just before he shoves Rocky and the match begins.
When Rocky is in training for the re-match with Clubber and is having flashbacks to the first fight, they cut to a scene in black and white where Rocky is taking an uppercut to the body. It is clearly visible that Rocky is wearing the "Stars and Stripes" trunks that Apollo gives to him right before the re-match. Rocky could not have been wearing these trunks because they hadn't even been given to him yet.
There are multiple errors with the time-line and chronology of this film and it's relationship with the previous two movies. Most obviously, Rocky is set primarily in 1975, with the title fight taking place on January 1, 1976. Rocky II is also set in 1976, with the title fight taking place on Thanksgiving, November 25. The events in Rocky III are said to take place three years later, placing it in 1979/1980. However, the newspaper advertising the charity match between Rocky and Thunderlips gives a date of 1982. Another major discrepancy is in relation to Micky's age and date of death. In the first film, Micky says he is 76, placing his birth in either 1899 or 1900. In Rocky III however, his headstone lists his date of birth as 1905, making him 71 in the original film. Furthermore, the headstone lists his date of death as 1981, which implies that he was 76; the same age he was four years previously (in the original film).
In the second fight with Clubber Lang, between the first and second rounds, the shot is flopped - the word "Clubber" is backwards on his shorts and one of his crew's robes clearly has "World Champion" backwards as well.
If you compare the movie time to actual time, during the third round of the final fight Clubber would have been knocked down at 2:56 of the third round. Therefore he would have been "saved by the bell" as the bell would have rung before he could be counted out.
When Apollo and Duke are reviewing Rocky's first fight with Clubber and discussing how many rounds Rocky could potentially last, the footage they are watching is in black and white. However, during this scene there is a shot of Apollo looking at the screen, and the lights reflecting off his face show that the footage he is watching is in color.
When Adrian and Rocky talk on the beach, in the first, zoomed-out shot of them together, there are numerous ships in the water along with some minor debris. In the close-up shots that follow, there are no ships, just the minor debris.
When Mickey gets sick and lays down on the table in the locker room, he has his knit cap on as he starts to lay down. The shot switches to a different angle as he continues to lay down, and now his cap is off and laying on the table next to him.
During the final bout, immediately before Rocky knocks out Lang, the position of the two fighters alternates from the middle of the ring (in long shots) to up against the ropes (in close-ups) and back again.
At the beginning of the final fight between Rocky and Clubber, we see Rocky take his robe off in a close-up shot and he has a towel around his neck. We then see a distant shot and the towel is gone. Then we come to a slightly closer shot and the towel is back again.
During the first round of the second fight with Clubber, Apollo is on the left, Paulie is next to him. When Apollo makes comments to Paulie ("He looks fantastic!") Paulie is seated to the left of Apollo, positions they do not assume until the second round.
Just after the rematch has started in one shot you see in the background Apollo is on the left and Paulie is on the right, then in the next when Apollo says, "He looks fantastic," Apollo is on the right and Paule is on the left.
The final fight supposedly takes place in New York City's Madison Square Garden; yet in the very first interior shot, a crowd panorama briefly shows the word "Philadelphia" along the concourse. The arena is actually the Philadelphia Spectrum.
At Micky's funeral, there are only 4 people present (Rocky, Adrien, Paulie and Al the Cutman. In order for a Jewish service/prayer to take place, there must be a minimum of 10 men (Minyan). If there is not a Minyan then the prayers have no validity and therefore strangers are often invited to the prayers ensuring there are 10 or more men.
During the first fight with Clubber while Mickey is lying on the table in the locker room, the doctor is overheard to say "his blood pressure is 200 over 50 we'd better start CPR." In actuality, the heart must be beating for there to be a blood pressure and CPR is only initiated when the heart and or respirations cease.
Mickey is revealed as Jewish in the funeral scene, but religious tradition requires a plain wood coffin with no metal be used (the coffin is obviously painted and clearly has metal handles). In addition, Jews cannot be embalmed, nor are they buried above ground. Fair enough to assume Rocky probably wouldn't know those facts, but not quite as fair to assume a rabbi wouldn't inform him.
As the scene depicting the dedication of the Rocky statue opens, the camera is focused on the snare drum of one of the high school band members. If one looks closely, it is clear that Mr. T autographed the top of the drum for the band member at some point before filming the scene.
(at around 6 mins) The newspaper clip showing Clubber Lang Win, the picture of Clubber Lang has a caption about a Mr. Danbury being charged with manslaughter with trial to begin on Tuesday. No article on the newspaper but the heading talks about the win.
During the statue dedication, the band plays, "Gonna Fly Now," marking the first and only time in the Rocky series that the song is played live where Rocky can actually hear it as a part of the actual plot. All other times, the song, along with the other soundtrack pieces play as a backdrop to the movie. Since backdrop songs to any movie aren't actually parts to the plot, the playing of the song, "Gonna Fly Now," would technically be a song Rocky has never heard before and he would have no idea what the connection is or why the song is being played at the dedication.
During the last fight scene, Rocky is punching away at Clubber, and the two seemingly glide across from one end of the ring to the other. If you listen carefully, you can hear the sound of the rolling platform that they are standing on.