Why did reanimated Itachi lose eye vision after using Izanami? As I understand, any physical part of a reanimated person cannot be destroyed. It will be regenerated automatically within a short time. Then, why did Itachi lose his eyesight after using Izanami? He should've got the eye back quickly as he was a reanimated person. Or, he could've damaged his own eye with a kunai, so that he would get a fresh Mangekyo Sharingan.
Q. Any physical part of a reanimated person cannot be destroyed.


The eye was not destroyed at all. It just lost its light forever. As stated in the wiki (emphasis mine)


  It is a genjutsu which affects the target through physical sensations shared between them and the user. Like its counterpart, in exchange for the temporary ability that it grants the user, the Sharingan with which Izanami is cast is rendered blind and loses its light forever.





  Q. It will be regenerated automatically within a short time.


Since the eye was never destroyed, it did not regenerate.




  Q. He could've damaged his own eye with a kunai, so that he would get a fresh Mangekyo Sharingan.


The light of the eye was already lost. Even if Itachi were to destroy it with his own kunai, the regenerated eye would still have been the one without its light.

Also, it should be noted that Itachi did not want to stay in the living world anymore, as he had no more attachments with the living (after telling the truth to Sasuke). Therefore, Itachi would never have tried it at all in the first place.