As far as we know, no casinos in Las Vegas have dedicated coat-checking facilities. It's not a bad idea, though. And it's not just in the winter when you can use a place to stash your extra clothing; during the summer, casinos set their air-conditioner thermostats on deep freeze and savvy patrons know to bring sweaters and light jackets in order to not to risk coming down with hypothermia — and this is when it's 115 degrees outside. But some places are warmer than others, so at times you might want to ditch the extra layers while you play or eat.
We didn't do an epic survey, because we've simply never seen a room in a local casino where an attendant takes your coat, hat, umbrella, or other outerwear and hands you back a claim check. Even many nightclubs (e.g., Rain, Ra, and Club Rio) don't have coat-check facilities, while those that do tend to charge you for the privilege ($2 per coat at rumjungle and $5 at Light, for example).
However, there is a play: Providing that its storage room isn't too full of baggage, you can usually check your coat in at the bell desk. We called a few casinos at random (Bally's, Palms, Barbary Coast, and Plaza) and all confirmed that they're happy to take your coat as long as they have space. They'll give you a claim check, just as a cloakroom would, and you don't have to be a guest of the hotel.
Another alternative is to valet park your car (if you have one) and leave your outerwear in the trunk. If it's especially cold when you pick up your car, you can wait in the lobby until it shows up.
