After the Battle of Shrewsbury, King Henry IV (John Gielgud) talks with his counsellors about Prince Hal, and from the beginning of the scene he wears a black skullcap. He says, "'Tis seldom when the bee doth leave her comb in the dead carrion", pushes the crown off a chest where it is sitting, and begins to fall. When he collapses to the ground, however, the skullcap has vanished, and Henry IV remains bareheaded for the remainder of the scene.
The knights are depicted being lowered onto their horses by rope, because their armor is so heavy. In reality this never happened and is not recorded in any text; the myth originated in the 19th century. A suit of armor only weighed about 50 pounds (20 kg) and knights could mount their own horses quite easily.
The corpse of Hotspur opens and closes his mouth several minutes after his death.
During Prince Hal/Henry V's speech immediately following Henry IV's death, a very obvious double for Sir John Gielgud lies slumped on the throne. (Scenes were shot out of sequence, and Gielgud was unavailable for that particular scene)
In the shot immediately following the goof where the corpse of Hotspur momentarily opens and closes his mouth, a dummy has clearly replaced the actor playing Hotspur. This particular take was filmed when he wasn't available.