When the Bolton archers draw and shoot at the Bolton-Stark cavalry battle, the Bolton cavalry can still be seen behind the archers. They are supposed to be on the field fighting.
When Ramsay retreats to Winterfell, he appears to have a dagger with him on his lower back. However, this dagger is missing when Jon confronts him inside Winterfell.
When Jon and Ramsay first meet to discuss when to do battle, Sansa leaves before Ramsay says he hasn't fed his hounds in seven days. At the end of the episode, Sansa says to Ramsay referring to his hounds, "You haven't fed them in seven days, you said it yourself." Sansa was never there to know that information, even though it was crucial in the manner of his death.
When Jon is climbing up to get to Ramsay, Wun Wun is seen very far away. But when Jon is on top, Wun Wun is right next to him.
When Jon rides his horse at Ramsay following Rickon's death, the Bolton archers fire their first salvo just as he does so. However, the wide shot of Jon's charge shows the battlefield already littered with dozens of arrows several seconds before this salvo is shown striking Rickon's corpse and the surrounding soil.
The small boats in the bay could not launch projectiles of the type shown the heights indicated. Such tension would tear the boat apart.
Jon's sword is visibly a prop when he mounts up to ride and save Rickon.
When wun-wun is shot down for a brief moment, an arrow can be seen around his right head floating in the air.
When Daenerys is about to mount Drogon (in order to attack the ships), a crew member can be seen walking from right to left, behind Drogon's right leg.
During the war council meeting, Tormund describes how Stannis cut through the Wildling army "like piss through snow." Tormund was not present when Stannis attacked the Wildling camp, as he was being held prisoner at Castle Black at the time.
When shooting arrows at Rickon, the time they travel in the air is long enough for Rickon to see where they're going and evade them. There is no need for him to run in fear of getting shot.
After the battle, Jon orders that Rickon's corpse will be buried in the crypts by their father's bones. Ned's bones were last seen and mentioned in Garden of Bones (2012), being presented to Catelyn by Littlefinger, and their subsequent fate was unknown. In fact, Littlefinger does not inform Jon that he had Ned's bones buried in the Winterfell crypts until Stormborn (2017).
During Jon's war council after meeting with Ramsay, Tormund asks if the Bolton army will be able to ride through them like Stannis did to Mance after the Battle of Castle Black. Tormund was in a cell in Castle Black when Stannis enveloped the wildling Army in a pincer movement, and wouldn't know what had happened.
At Hardhome, Wun Wun the giant was enthusiastically wielding a large flaming pole against the ice wights, showing that even if he has a preference for smashing and ripping with his bare hands, he will use at least a simple weapon when he really needs to. Before this battle started, he could surely see that his side was vastly outnumbered by the enemies, so he should have armed himself then - there were plenty of trees nearby that he could have uprooted and stripped the branches from. He does eventually start using a large shield taken from the enemies to scoop them out of the way, but only after he's been stabbed many times.
Before battle, Ramsay says that he has not fed his hounds for seven days. One day later after battle, Sansa repeats him the same thing, but it should be eight days.
Tyrion uses the phrase "by the grace of Her Majesty" - up to this point, all monarchs have been referred to as "Your Grace," much as they were in real-life England up to and beyond the time of Henry VIII. Presumably this was to avoid the repetitious phrase "by the grace of Her Grace."