Deliberate mistake: When the two Confederate and Union soldiers meet across the river and trade coffee/tobacco, you can see the other side of the bank behind them. It looks at least 50 feet away, yet before they were able to talk to each other without shouting. Also, the Union soldier raises the pipe to his mouth in his right hand, which switches to his left in the next shot. There is also a mist present at the beginning and end of the meeting, but not in the middle.
Deliberate mistake: At Fredricksburg, when the troops from the 20th Maine are retreating, Tom Chamberlain is seen firing his revolver at the Confederates, and there is no burst of smoke, nor any recoil (also, the tip of the barrel of the pistol is off-screen.) This is because he is dry-firing the gun, since that shot contains dialogue and a real shot would interfere with the microphones. This mistake occurs earlier as well, with the Irish Confederate officer firing his revolver at the Irish Union soldiers, though no bullets are visible in the chambers, and the smoke from the primer caps is missing.