Do Akatsuki rings have abilities? Each of Akatsuki's members wear unique rings.
I guess Itachi's one can use genjutsu. 



But how about others? Can it be used as ninjutsu or genjutsu abilities?
And why does Zetsu always collect those rings after each one of the members died.
The rings do not have abilities. 

They are simply used for the Akatsuki membership, which is why Sasuke's group Taka was never really considered Akatsuki. They might have had the cloaks, but they didn't have the rings and the rings signify a true Akatsuki member. 

Also, Orochimaru's ring was never replaced when he left the Akatsuki. There have been 10 Akatsuki rings total as well as 10 members of the group.

Taken from Naruto wikia 


  Right thumb: "zero" (零, rei); worn by Pain. Its colour is purplish-grey.
  
  Right index finger: "blue," "green" (青, ao, shō); worn by Deidara. Its colour is teal.
  
  Right middle finger: "white" (白, bya); worn by Konan. Its colour is white.
  
  Right ring finger: "vermilion," "scarlet" (朱, shu); worn by Itachi Uchiha. Its colour is >red.
  
  Right little finger: "sign of the boar" (亥, gai); worn by Zetsu. Its colour is green.
  
  Left little finger: "sky," "void" (空, kū); worn by Orochimaru. Its colour is slate blue.
  
  Left ring finger: "south" (南, nan); worn by Kisame Hoshigaki. Its colour is yellow.
  
  Left middle finger: "north" (北, hoku); worn by Kakuzu. Its colour is dark green.
  
  Left index finger: "three" (三, san); worn by Hidan. Its colour is orange.
  
  Left thumb: "jewel," "ball," also the black king in shogi (玉, gyoku); worn by Sasori and >later by Tobi. Its colour is purple.