My ice cream cone is melting.
Analysed literally, this is not sensible as katie_here said.
Mark1978 suggests that "everyone knows" you mean the ice cream is melting.
That's true enough, but relying on everyone knowing what you mean even though you say something quite different is not a good approach to communication.
In the real world, I would be very surprised if anyone actually said "My ice cream cone is melting." The sentence without the cone is a lot more natural.