Could you feed my dog when I'm gone? really does sound as though the person speaking will be disappear permanently. We often use the phrase 'when (a person) is gone' to mean when they are dead. Could you feed my dog while I'm gone? sounds slightly less permanent. If you are just going to be 'gone' for a short time, it might be more natural to say Could you feed my dog while I'm away?

Both are fine. The first has more a sense that after leaving, you won't return, maybe because of death or moving away. The second is more general, used when you will be gone for a definite period of time.