Before Hallie is piercing Annie's ear, Annie is holding an ice cube behind her ear. The ice cube doesn't melt at all. Annie's hand isn't wet, neither is the bed or anything else. It's obviously a fake one.
OK, this IS the movie business! Ice may not melt that fast, anyway. If I had a piece of ice in my hand, it wouldn't completely turn into water in a second!
Actually, you can see the ice melt. A very small amount of water is on Annie's fingers. The scene went by fairly quick and did not have enough time for the whole thing to melt.
Well if they used a real ice cube, if during filming, one of them laughed, they would have to start again, and the ice cube would have melted by then. And anyway, who would want to freeze their hand?
The studio lighting they use on the sets create so much heat (probably around 90 degrees by themselves) that anything frozen would melt in a matter of minutes. If a real cube was used, there would be water everywhere, and the cube would have to be replaced. If shots had to be re-taken, the cube would seem to magically grow and shrink as the camera angle changed.
As far as the characters in the movie go, just where would they have gotten ice in an isolated cabin in the first place?