While I'm sure a real tech will come by and give you his opinion, I'm willing to bet that it'll start with, "It depends ..." Number one thing that it'd depend on would be the material your clarinet is made out of: wood, metal, plastic, or hard rubber. Wood can dry out and could crack, which isn't a good thing. The other materials don't really have that problem. Answering the specific question of "cork sticking," the answer is "yes." I've had a couple instruments where that happened. However, they might have been stuck together for years, so time will make a difference. Same could be said for pads drying out and falling off. Or molding.
You didn't mention one obvious thing: if the horn's left out, it's more prone to being accidentally broken. You really don't want to hear, "Ooops. Just tripped over your clarinet."
You should disassemble the horn, clean it with a swab, and put it in its case. It doesn't take that long.