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.
At the end of Rocky IV, 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.
Near the very end of the film, when Rocky confronts George Washington Duke, after Duke says, "Touch me and I'll sue," we see a quick close-up of Rocky clenching his left fist. Then when Rocky is about to throw the punch he drops his right shoulder as if he is going to throw a right-handed punch. However, when the angle changes we see Rocky throwing a left-handed punch.
When Paulie is playing Santa Clause and has his "laundry bag" in his right hand he grabs Rocky Jr. with his left hand asking him if he's been nice then when the camera angle changes Paulie is grabbing him with his right hand.
When Rocky has his first press conference at the airport, after returning from Russia, he is approached by Duke who is forcefully challenging him to fight Cane. At one point Adrian stands to protest and after an immediate retort from Rocky and Duke the camera shows a wide shot and Adrian in still in her seat beside the podium.
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.
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 (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.
The goof items below may give away important plot points.
Paulie admits to Adrian that he signed over Power of Attorney to Rocky's accountant while Rocky was in the Soviet Union, resulting in Rocky losing his fortune. However, Rocky's lawyer states that the accountant gave Paulie the Power of Attorney form for Rocky to sign before Rocky left. Neither statement can be true.
Rocky lost his fortune because Paulie signed a paper giving Power of Attorney to Rocky's accountant, who promptly used Rocky's money to invest in several shady ventures. The accountant had Paulie sign because Rocky would be in the Soviet Union. But Paulie was in the Soviet Union with Rocky, so he couldn't have signed over Power of Attorney.