Character mistake: On the handwritten label Bond finds, Bratislava is written Bratislavia.
Factual error: Both in Czech and in Slovak, the forms of last names always mark gender (aside from some exceptions which aren't relevant here) 'Milovy' is not a possible last name for a Czech or Slovak woman because it is a male adjective. 'Milová' would be possible, though. This kind of mistake is very commonly made with Slavic names (e.g. Russian women whose last names end in '-sky' when they should end in '-skaya') Also her first name 'Kara' is virtually never used in the Czech Republic or Slovakia (let alone communist Czechoslovakia) and sounds very exotic to the people that live there. We can just assume she got an unusual first name for some reason which is entirely possible, but overall her full name sounds very silly to a native speaker.
Continuity mistake: When the plane crashes at the end all four engines are running at high speed, but before that all four engines had stopped because they ran out of fuel.
Continuity mistake: When the stenographer is taking Koskov's account of his escape, there is no paper in the tray and the stenographer isn't pressing the keys. Minutes later the agent/milkman enters and removes paper from the machine.
Factual error: While Bond is taxiing down the runway at the Russian airbase, he is nowhere near takeoff speed for that aircraft (if he was, the Jeep wouldn't be able to catch up).
Continuity mistake: In the teaser, there is something wrong with the landing of the second agent. He lands on a dirt road and runs to his left behind some bushes while pulling his parachute towards the ground. To his right is a rock wall. In the next shot he lands again - presumably below that rock wall. But he was running in the opposite direction. If he changed direction and jumped over the wall to the right he would have run into his parachute. And it can't be a wall behind the bushes to the left he jumped over, because you can see from the shadows that the sun comes from the low side of the wall in both shots.
Continuity mistake: When his plane runs out of fuel, Bond and Cara eject in the jeep. When the jeep ejects, the plane is just a couple meters off the ground. Cut to a shot of the plane crashing into a cliff, it's at least 10 to 20 meters up. Impossible for the plane to have gained altitude; the engines were off and no updraft of air would be strong enough to lift a plane that size.
Audio problem: At the fairground, James tells Cara that he has to check a message, but his mouth doesn't move when he says this.
Continuity mistake: When the car cuts the large hole in the ice Bond doesn't move when he finishes the cutting the circle, but we see the view from his car, and he's several feet away. Also the size of the circle he cut out dramatically changes to being a lot smaller.
Factual error: When James Bond and Kara Milovy are on the plane and it runs out of fuel, one engine quits first. When this happens, the aircraft turns towards the stalled engine, so the pilot has to compensate by steering towards the good one, or the plane will spiral. James Bond doesn't do this, he just keeps the wheel straight and pilots normally. (01:59:17)
Continuity mistake: When Bond stops the timer on the bomb in the transport plane, the digital readout shows 3 seconds to detonation. When he restarts the timer to explode the bomb on the bridge below, the time to detonation has rather conveniently changed to 9 seconds. This gives the bomb time to land on the deck before it explodes.
Continuity mistake: When Bond and Milova escape Bratislava it is snowy and wintry. When they arrive in Vienna, which is only 50 miles away and at the same latitude, it is late summer or early autumn, with signs of leaves changing.
Audio problem: When Saunders says "In the box, between the KGB minders" at the concert, his mouth is not moving.
Factual error: When Bond steals the opium laden plane from the airbase, Kamran Shah and his men are busy fighting the Russians. Considerable flying time later (during which Necros gets the boot), Bond then drops the bomb on the bridge. Somehow Kamran has managed to get ahead of the plane on horseback.
Continuity mistake: When Bond exits the tunnel in the Aston Martin they get fired upon by an armoured car - look at the scenery outside the Aston, from inside there is no left turn, but one appears out of nowhere for Bond to take and head down to the frozen lake.