Salt and snow have a complicated relationship, and yes — under the right conditions salt can make a snowman’s carrot nose go sad overnight, but it’s not as cinematic as you might hope.
Chemistry first: salt (sodium chloride) lowers the freezing point of water, so when it contacts snow it turns the surface into a salty brine that melts at temperatures where pure ice would stay solid. In real-world terms, common road salt works well down to around -8 to -10°C; below that it becomes far less effective unless you use something like calcium chloride which works at much
colder temps. A carrot is mostly water, so if a decent amount of salt gets onto or around the nose, that brine will soften the surrounding snow and can make the nose lose support or start to sit in liquid instead of firm snow.
Practically, a single pinch of table salt dusted on the tip probably won’t make the whole carrot disappear by morning. A handful of salt or salt spread over the area, warm daytime sun, wind, and temperatures hovering near freezing — those combined will do it. Also, salt pulls moisture from the carrot’s cells (osmotic action), which can leave the nose shriveled and mushy even if it doesn’t fully dissolve. I’ve watched a snowman’s face collapse after a neighbor salted the path — not dramatic melting, but a slow, soggy surrender that left the carrot looking sad by sunrise.