The Ilex shrine is found generated rarely in Forest biomes, it is used to spawn Celebi and alien Celebi.
When you right click the shrine with the corresponding ball it will spawn the Celebi into battle with the player
To spawn a regular Celebi you right click the shrine with a GS Ball in hand.
To spawn an Alien Celebi you right click the shrine with a Beast Ball in hand.
Like the Trio bird shrines the Ilex shrine can't be mined.