This is a very interesting question. I've never had an answer to it, so i generally cut them off once they start to rot.
I've left a few on before, they are rather odd. The lid and the pitcher down to the top of the digestive shrivvels up, but the digestive section at the bottom doesnt. I've noticed this on ones I've cut off too, they go a gold colour eventually.
Almost like the plant either keeps the lower section to finish digesting the prey, or may be the digestive enzymes destroy and microbes which would normally cause the rotting process. They do rot eventually, just takes awhile.
So in answer to your question, I dont think it really matters, if the pitcher is empty, may as well cut if off, but if theres something in their could leave it.
Personal choice until otherwise
