Why didn't Obito die after using Rinne rebirth?
In Chapter 657, right after Madara is brought back to life using the Rinne Rebirth jutsu, the scene moves to the Kamui dimension, where Zetsu confronts Kakashi, Minato, and a fatigued Obito. 

Zetsu explicitly stated that Obito will die from using the jutsu. And then,


  


So Black Zetsu was the reason Obito was able to postpone his unavoidable death.