The speaker spends the first stanza of "Funeral Blues" jawing about how much he wants everyone and everything to pipe down. Maybe he wants some peace and quiet to deal with his thoughts. Maybe he wants to make sure that everyone can hear his lament. Maybe he wants silence out of respect for the dead man. Whatever the answer may be, he sure does a bunch of talking about not talking.
Line 1: The speaker wants to cut off personal communication with the world: he wants to stop the telephone lines from running. He's looking for isolation. He's probably being hyperbolic here, which means that he's exaggerating his feelings and desires to show just how sad and hopeless he is.
Line 2: He also wants to stop dogs from barking. Poor dogs. It's not their fault.
Line 3: Now he'd like people to quit playing the piano, thank you very much. Seems fair enough. This is a funeral after all.
Lines 3-4: He wants to hear the "muffled drum" of the funeral march. The speaker wants to hear this and this only. It's like all other noise is a distraction from what really matters, which is his pain.