Homer is wearing his green backpack when he and Burns get on the snowmobile to race to the cabin first, but it disappears when they get there.
The order of the drills/alarms on Burns's desk changes between long shots and closeups.
When Homer opens the cabin door after the first avalanche, the door opens with hinges on the left. After the second avalanche, when Burns opens the door, the hinges are on the right. When the cabin arrives at the bottom of the mountain, the hinges are back on the left (when viewed from the inside).
Earlier Smithers tells Mr. Burns he doesn't know how long the fire drill evacuation lasted because his timing watch only goes to fifteen minutes, yet later at the end of the episode long after forty-five minutes in-verse, Smithers tells Mr. Burns they've set a new record, using his timing watch to check.
After Homer and Mr Burns end up back in the cabin following their failed escape attempt, the slightest noise they make causes further avalanches, but their normal conversation earlier didn't.