Will Sasuke's Kirin work against Obito?
We don't have any proof for the moment that it would work or not against Obito.

But I would say that Kirin wouldn't work. It's categorized as a Raiton jutsu. So it's still ninjutsu and Obito should not be damaged.

Also, to use this jutsu, Sasuke needs to use a Katon jutsu to create the clouds. So, when the clouds are charged with thunders, it may still have some energy of the previous, meaning that it's not really natural lightning.