This song is a masterpiece. The sentimental tune, borrowed from The Parting Glass, amplifies the emotional power of the lyrics, which are pure poetry.
The song is about a life lived honestly and the life beyond this life that waits. He's lived honestly and will leave this world without giving a damn. It's only a "false" clock that ticks out his time -- just an annoyance to distract him. The dirt and dust that covers his face (when he's buried) are merely "rumors" and "gossip," easily pierced by the arrow of the truth. It's the darkness that dies when the curtains are drawn (death) and the dawn that follows.
Note: The lyrics on Dylan's website have an error. It says "The time aint' tall, yet on time you depend." The lyric on the album "The time ain't tall, if on time you depend." The "if" conveys the opposite meaning of "yet" and is consistent with the interpretation that it is a false clock that counts down our days.