If Obito is able to isolate which body parts can be phased into the Kamui world, he can isolate his body off of the flames. After all, his arm is in contact with his shoulder, his shoulder with his chest, his chest with his neck and sternum, you get the idea, and yet, those don't have to go with him. So, neither do the flames.
Two sets of Konan's attacks hit Obito. The first hit because Obito was allowing the paper to pass through his body, then solidified to grab Konan. He attempted to suck up Konan, which rendered him solid. As he did, she detonated the tags, which blew off the arm he was holding with her with and damaged that half of his body, but not his entire body because most of the explosion was sucked up.
The second time, it was an attack that would have lasted for ten straight minutes, and Obito can only go intangible for 5 straight minutes. So, he had to use Izanagi.
Neither instances were because he couldn't let them phase through him, but because Konan exploited moments where he was tangible.
