Other mistake: During the scene in the banquet hall, as Tormund says "most people get bloody murdered, they stay that way", look at Daenerys in the background - a disposable coffee cup is on the table in front of her. This brown cup with a white plastic lid is nothing remotely like the style of cups they're all drinking from in the scene. HBO acknowledged the slip up and digitally edited it out on streaming services, reruns and future DVD releases.
Character mistake: Gendry gets declared as Lord Gendry Baratheon of Storm's End by Daenerys. But he refers to himself as Gendry Rivers - the surname given to bastards from the Riverlands. Gendry comes from King's Landing, and bastards from there are called Waters, not Rivers. Given that Gendry is not an acknowledged highborn bastard, he should never have had a surname in the first place.
Other mistake: The dragon is shot down by three giant arrows, and everyone looks shocked, with no idea where the attack came from, even Daenerys, who's flying high alongside it. Then the enemy fleet rounds a small outcrop, revealing themselves as the attackers, but given her height she would have seen them a mile off and steered well clear.
Continuity mistake: When the episode starts Arya is walking towards where Daenerys is and has her face covered in mud and soot, plus a long streak of dark dry blood running through the right side of her face. The next time we see her her face is cleaner and the blood is less noticeable. In further shots the amount of blood increases or decreases randomly. There doesn't seem to be much water and soap around the place.
Visible crew/equipment: After Sansa tells young Ned Umber to hurry back to Last Hearth and bring his people to Winterfell, when Jon tells Maester Wolkan to send ravens to the Night's Watch, in the next shot of Wolkan the chalk T-mark can be seen on the stone floor in front of his feet (it's not the stone's edges or cracks). (00:11:20)
Character mistake: When Renly and Catelyn are negotiating the terms of Renly's alliance with Robb, Renly calls Moat Cailin "Moat Caitlin." As someone who served as Master of Laws on King Robert's Small Council, he should know the correct name of the territory he is referring to.
Continuity mistake: During the meeting at the Dragonpit, Sansa is wearing her long silver chain and circle neckwear, with the long chain changing the way it is placed through the circle. First the chain is put through the circle in a way that's over and under, but in some shots the chain is put through the circle the opposite way, under and over. (00:53:10)
Visible crew/equipment: When Bran joins Tyrion and the others at the small council meeting, Bran begins by saying that they appear to be missing a Master of Whisperers, then just as he says, "And a Master of Laws," the boom mic can be seen dipping into view over Bran, at the top of the screen (visible on Hulu). (01:11:00)