I figured I'd actually try posting an idea that I've heard around the base, but add some ideas to it.
I think this is most likely a set up. Obito told Zetsu about Kakashi and Rin, but that picture of their team was when he was still in Konoha, right? Obito mentioning Rin and Kakashi shouldn't have mattered, yet as soon as Zetsu returned, he said Rin and Kakashi were in danger. The fact that the two of them were in danger shouldn't have even been important to the Zetsu, since he barely knew much about them.
From what we know there was a dozen or more Mist ninjas surrounding Kakashi and Rin. Rin was a medical ninja, so she wouldn't leap at the Mist ninja like Kakashi would. Rin would try to support Kakashi as a medic, so it could have mainly been Kakashi fighting against the Mist ninja. Going up against that many ninja, even if Rin was helping a lot, what would the possibility of them both surviving until Obito arrived be? I'm not sure how far Obito had to travel, but besides Kakashi hurting Rin, shouldn't the both of them had at least some noticeable injuries?
That's why I think it's a set up. I'm not going to say Kakashi killed her, since Obito said Kakashi let her die, not killed her. Another theory I have is a clone or transformation, even though it is not likely at all. I only saw that because on the last page we can see the sign of a battlefield in the back. Mainly smoke. With all that smoke there is always a chance of a ninja using transformation to make it appear that Kakashi harmed Rin.
The possibility of him doing it on purpose is there as well, but I haven't come up with much reasons as to why he would do it on purpose. From the shock and reaction from Rin, I don't think she was a spy for the Mist ninja.