The speaker of "Funeral Blues" wants us to put out the stars and dismantle the sun. These hyperbolic statements and the ones that follow are all about shutting down the natural world in order to demonstrate this poor guy's grief. It seems like the speaker knows that his commands are hyperbolic, exaggerated, and impossible, but thinks that nothing smaller than nature itself can communicate his despair accurately.
Nature is universal. The speaker issues his commands against nature so that everyone can understand his pain.
The speaker doesn't want nature to "end" for everyone. He just doesn't want to be reminded of the beautiful and romantic world out there while he's grieving.