The only two champions I've seen use Doran's successfully jungling are Aatrox and Shaco. I don't understand how it works on Shaco -- I did it once on accident and only did well because of a lucky start. On Aatrox, it works because he has insane built-in sustain with his W. Aatrox literally never HAS to b when jungling and maxing W, you only b because you choose to for item purposes (or post-gank/fight to be safe -- because his W heals double under 50% hp, even if you're really low as Aatrox, you can still farm the jungle with a doran's!)
There are surely other champions that can Doran's start, but I can't think of them off the top of my head. Aatrox not only can start Doran's, but he SHOULD.