I have curtain rod holder which I am not able to remove from wall. Netiher it rotates nor i can pull it out. I don't want to use any pressure too much because I don't want to damage the tiles.
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Well it appears to either be a threaded stud or screwed in from behind the tile.
I would gently try 2 flat head screw drivers in the slots and see if I could get the center to turn. If its a threaded stud you will be able to grab the stud with pliers and it will twist out. If its screwed in from the other side (really hard to believe that it would be but I've seen stupider) your gonna have trouble getting the end of the screw to go away.
This type are usually only held by friction, so the contact should only be synthetic rubber.
Try to cut behind it with a utility knife or a wire.
the socket that holds the end of the rod appears to be held in place with a large screw, you could try unscrewing the socket and see if the screw is attached to the wall or to the contact pad. and see if with the socket removed any further fasteners are revealed.