In Chapter 691, after Obito died, Kakashi’s Sharingan faded. Why is it that Kakashi's Sharingan faded, while the Sharingan Danzo had stolen from Shisui and the one Shisui had given Itachi, were still okay even after its original owner had died?
Both Kakashi and Danzou had physically obtained their Sharingan via transplantation.
During the finale of the Fourth Shinobi World War, Madara rips Kakashi's Sharingan and transplants as his left eye (Chapter 674).
Naruto, using Hagaromo's power, creates a normal eye for Kakashi (Chapter 675).
From Naruto Wikia article on Kakashi Hatake, merged plot summary of Chapter 686, 687 and 688 (emphasis mine):
Prepared to give his life to lend what little assistance he can, Kakashi stands in front of Sasuke to take the attack while Obito does the same for Naruto. Although they are unified in their willingness to die, Obito decides it's too soon for Kakashi and uses Kamui on the attack bound for him, leaving Obito unable to save himself. Obito's body dissolves, but his spirit returns shortly afterwards and inhabits Kakashi. Obito feels that Kakashi will become the next Hokage and wishes to reward him: a temporary moment of having two Mangekyō Sharingan.
Obito gave Kakashi the privilege of having both Sharingan (Mangekyou) temporarily. This was the doing of the spirit of Obito and not the physical self. Hence the Sharingan Kakashi used was spiritually influenced.
Later, when Kaguya is defeated, Obito's spirit leaves Kakashi's side to join the other world. Because of the spirit leaving, the Sharingan also disappears along with it.
Had the Sharingan been physically present, it would have stayed, just like Danzou, the Kotoamatsukami crow and pre-war Kakashi.
Sharingan that have been transplanted unto someone will never fade, even if the original user dies, as was the case with Danzo/Shisui. The original Sharingan that Kakashi had recieved wouldn't have faded either.
However, in chapter 674, Madara steals Kakashi's Sharingan. Kakashi doesn't regain the Sharingan afterwards.
In chapter 686, after Obito sacrifices himself to save both Naruto and Sasuke, his body dies, but his spirit is still alive and it inhabits Kakashi, leading him to gain two Mangekyo Sharingan.
After Kaguya's incapacitation, as Obito's spirit disperses, so does the Sharingan.
Basically, Obito dies when he attempts to save Naruto and Sasuke, but his spirit was still there, even though his original(physical) body had left the world. Now the spirit Obito grants Kakashi 2 spirit MS, which disappear after Kaguya was sealed away and Obito's spirit leaves the world. If Obito were to give kakashi his MS while he was alive, Kakashi could have kept the Sharingans even after the war. As for the original Sharingan Kakashi had, Madara snatched it out of Kakashi's eye socket(gross)...