At the end of Rocky IV (1985), Rocky's son (born in 1976) is 9. However, at the beginning of Rocky V, which in the movie's time-line takes place only a week after the end of Rocky IV, the boy is already 12 or 13. In addition, Rocky IV is set in 1985 (we know this because it is nearly the 9-year anniversary of Rocky and Adrian, and they were married in 1976), yet in this movie, Rocky Jr. and his friends are watching the 1989 movie National Lampoon's Christmas Vacation (1989).
During Rocky's fight with Drago in Rocky 4, Duke's mustache was black. In the beginning of this movie, which takes place in the locker room after the fight, his mustache is noticeably gray.
The house Rocky leaves to fight Drago in Russia is clearly not the same house he returns to in Rocky 5.
Even though the events in this movie are supposed to be directly after the fight in Rocky 4, Adrian's hair is about 5 or 6 inches longer than during the fight.
The beginning of the movie is supposed to take place right after the end of the previous film, however, Adrian's hair is noticeably longer and straighter.
During the end credit montage - which shows stills from all five Rocky movies chronologically - the still of Rocky running through the Italian Market during the first film's section is actually a shot from Rocky II (1979) (in which he also ran through the market).
When Rocky is having a flashback of training with Mickey when he is in the gym Mickey gives him the necklace that he said was given to him by Rocky Marciano. However, in Rocky II Apollo is wearing the same necklace when he crumbles Rocky's picture off the mirror. However, in this exchange Rocky asks "what happened to his other cuff link?" and Mickey replies "I don't know. He only give me one. He gave it to some bum." In all likelihood, the person that got the other cuff link was Apollo Creed.
At the press conference at the airport when Rocky returns to the states he is challenged by Union Cane for the title. Rocky gave up the title in Rocky IV to fight Drago in Russia, therefore he was not the champion, so for Union Cane to challenge him would have just been a regular fight not a championship one. However, it is never stated that Rocky is the current sanctioned champion. When Rocky left for Russia the title became vacant, he and Union Cane could potentially be fighting for the currently vacant title providing this was approved. This is exactly what happens when Cane eventually wins against another opponent.
Rocky's championship belt is in the basement despite burying it with Apollo Creed's body in Rocky IV. However, as both men held the championship it is likely that both would have had replicas that they could permanently keep.
When Rocky arrives back in the US from Russia, he is seen getting off an Ilyushin IL-86 jet. Assuming he arrived back in Philadelphia from Moscow (a distance of just over 4,000 miles) it would be impossible to fly that distance on an IL-86, which has a maximum range of just over 3,000 miles. It is never stated in the film that the flight back from Moscow was a nonstop flight, though.
The large house Rocky is forced to leave is not the same house as in Rocky IV and III. However many years pass between Rocky becoming rich and losing his fortune, so it is entirely possible the family moved at some point during the height of Rocky's popularity.
When Tommy Gunn is delivering one of his punches in the big fight, the Adidas symbol on his shirt is on the wrong side, revealing that it's a mirrored image.
At about 18m20s, there is a shot of a newspaper with the title 'Rocky's Bankrupt', the text underneath is completely unrelated to this.
When Rocky and his family move back to the old house and everyone is crowding the street, after Rocky Jr. places the box on top of his head and walks away, a camera man is visible in the window on the top floor of the house. The red blinking light on the camera is clearly visible.
When Rocky Jr. is arguing with his dad about Christmas, Rocky tells him to remember last year, even though last Christmas he was fighting in Russia and was not with his son then either.
After the fight is over, Rocky makes the statement that (paraphrased) he is sure that his kid has been watching the fight even though he should have already been in bed. With the time difference in Moscow, even if the fight had finished in Russia by mid-night then it would have been 6 p.m., or early evening in Philly.
In the beginning of the movie, in the shower, Rocky is talking about something that Mickey said about when he was fighting, right before he died. However, right before Mickey died in Rocky III, he didn't say any of that.
In the flashback scene, Mickey gives Rocky the boxing glove necklace before his fight with Apollo. However, when in the basement, he tells Tommy that Mickey gave it to him "before he departed". Mickey died about 3 years after Rocky's last fight with Apollo.